Differential Geometry of Three Dimensions [I]

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Citation preview








London Office Bentley House, N.w. I Amenoa.n :Bre.nch New York Agents for Cane.d&, Indla, e.nd PaJrlste.n • Ma.omtll&n

First Editlon 1027 Ropnnted 1031 1939 1947 1955

E1,rat printed 1,n Great Bntam at TJUJ Un,verauy Pre11a, Oa,mbndge B ~ by SpotturwoofU, llallan,yne &, Oo,, Ltd , Ookhuter



HE present 1mprese1on 1e eubstantieJly a repnnt of the origme.l work. Smee the book we.e firet pubhehed e. few errors have been corrected, and one or two pare.graphs rewritten. Among the fnende and correspondents who Iandly drew my attention to desirable changes were Mr A S. Ramsey of Magda.Je,,e College, Cambridge, who suggested the reV1S1on of§ 5, o.nd the le.ta R J. A Barnard of Melbourne Umvers1ty, whose mfluence was partly respollSlb!e for my initaal mterest m the subJect. The demand for the book, since its first appee.re.noe twenty years ago, has juetafied the writer's behef m the need for such a vector1eJ treatment. By the use of vector methods the presentation of the subJect IS both simphfied and condensed, and students are encouraged to ree.eon geometnceJ!y rather than aneJytaceJly. At a le.ter etage some of these students will proceed to the study of multid1mension11I differential geometry and the tensor calculus. It 1e lughly desirable that the etudy of the geometry of Euohdcnn 3-space should thue come first, and th1S can be undertaken with most students at an ee.rher stage by vector methods than by the R1cm ceJcnlue. A student's appremo.taon of the more general ea.ea will undoubtedly be enhanced by an earlier acquo.mte.noe with d1ffettntial geometry of three dimensions The more elementary pa.rte of the subJect are d1scuased in Chapters 1-XI. The remamder of the book 1e devoted to differentieJ mve.rie.nts for e. surface and their e.pplicationa. It w1ll be apparent to the ree.der that these oonetatute a powerful weapon for analysing the geometrical properties of eurfacee, and of eysteme of curves on a surface. The unit vector, n, norme.l to a surface at the current pomt, plays a promment pa.rt m this discusBion The first curvature of the surface 1e the negative of the divergence of n; whtle the second curvature ie expre&B1ble Bimply in terms of the divergence and the Laplucmn of n with respect to the eurfaee.



Extensive applications of these mva.rianta to the geometry of au,facee are given in the seoond volume of this book. Apphcations to phymcal problems oonnected with curved surfaces have been given elaewhe,e• by the author. • 1. On dd?erant11l 1nvana.nt& m geometry of 1tll'f&c,e11 ,nth 1ome appllaa.bona to ma.thema.tioal pb.Jal.01 Qua.7'terl)! Joumat of Machema.nc,, Vol a:o, pp. 290-69 (Oombddge, 19111). I On em.all deformation of n:rfa.oea a.nd of thm elastio ahella. Ibid., Vol. &01 pp. 971-06 (1996), 8. On me motion of an e:r.tenuble membrane m a. givan ourvecl aurfaoe, PAU M•g , Vol 98, pp 578-80 (1987), 4 On tra.naverae vibr&tl.ona of cur,ed mambra.nea. Plut Ma.g, Vol 28, pp 68184 (1989).


J. 1

P:mR'l'H1 W.IIIS'J!IIBN AD'B'l'BALU., llll Ja.MMM'!J, 1947,





.,.,_ 1. TtLngent •


2. Pnoc1p&l normal Ourva.tnre .


3. BiuormeJ. Tonuon. Serret-Frenet formul&a . 4 Loous of centre of ournture

13 17 18 21

EXA.>i.µ.' the ps.rametrio ourves a.re the generators. What curves a.re represented by X=,,., and by )..µ.=conet i

l33. First order ma.gnltudea. The suffix 1 will he used to indicate partial differentiation with respect to "• and the suffix 2 pe.rtial d1fferent1at10n with respect to 11. Thus r1

or =au,



r!il=av, ~r


ru=aui, ru1=auav' r-=av-· and so on. The vector r, 1s tangential to the curve 11 = conat at the pomt r, for its direct10n is that of the displacement dr due to a variation du m the first parameter only. We take the positive direct10n along the parametnc curve v = conat. a.s that for which u mcreases This is the direct10n of the vector r1 (Fig 10) Similarly r, 18 tangential to the curve U = const lll the positive sense, which corresponds to mcrease of v. Consider two neighhouring pomts on the surface, with position vectors r and r + dr, corresponding to the parameter values u, v and "+ du, v + dv respectively Then



dr = a,;, du+ a,j dv

= r,du + r,dv. Since the two points are adjaeent points on & curve passing through them, the length ds of the element of aro JOmmg them is equal to their actual distance Idr I apart. Thus ds' = dr• = (r1 du + r,dv)' = r,•du• + 2r1 •r,dud• + r,'dv'. If then we write E=r1~ F= r1 •r,, G=r,• ............. (1), we have the formula ds'=Edu'+ 2Fdudv+ Gdv' ............... (2). The qus.ntities denoted by E, F, G are cs.lled the fr,ndamental



magnitudea of the flrt!t {FJ:dv, and the umt tangent to this curve b


ar G-i r,. = ,,;Ga.= 1

The two pa.ra.metnc curves through any pmnt of the surface at an angle o, such that r1 .r11 F , COBOJ=O.•b= ,,/EG= ,JEG Therefore•

Blll0>=JEGE(/'=-;;-!0• . .......... .. (4;


H tan"'= F

Also since 1t follows th&t



= la. xb I= ,JE(J.I r, x r,I, I r, x r,I =H ........................ (5

The parametric curves will cut at right angles at any po:i.:i: F = 0 at that point, and they wtll do so at all points if F = O < the surface. In this case they a.re said to be , •=f(u), the longitude rp being the inclination of the we.I plane through the given point to the sa:-plane. The pa.rametric curves •=constant are the • mendian lines," or rntereectione of the surface by the axial planes; the curves u = constant are the • pa.rallels," or intersections of the surface by planes perpendicular to the e.x1e With u, , ucoerp, 0). The first order magnitudes are therefore

E=l+f,', F=0, G=~, H•=u'(l+f,'), S1noe F = 0 it follows that the parallels cut the menchane orthogonally. The unit normal to the surface is n=(-f,ucoe,i,, -/,um,i,, u)/H. Further ru = (0, 0, /u), r,.=(- sm rp, cos, 0), r,.=(-ucoerp, -usm,i,, 0), so that the second order magmtudee are

L=uf,,/H, M=0, N=u'fJH, T'=u•fJ;.,/H•. Smee F and M both vameh identically, the pa.rametnc cunies II/I'll

the lines of ouroa.ture.




The equat10n for the pnnc1pa.l ourva.turea reduces to

u(l + f.')'i(u,v)= const. are 11a1d to form a wnjugats 6 w.




system._ At a point of intersect10n of two curves, one from each family, their directions a.re conjugate. Further, given two families of curves cf> (u, v) = const., u, v) = const., we may determine the condition that they form a conjugate system For, the directions of the two curves through a pomt u, v are giveIJ by


cf>iBu+ q,,ov=O} v,du+v,dv=O '

It then follows from (I 7') that these directions will be conjugat, if Liv,+ c/>,t,) + Ncf>,t, = 0 ..... (20). This ia the neceaaary and sufficient condition that the two familie: of curves form a conJugate system. In particular the parametric curves•= canst., 11, = canst. will form a conjugate system if M = 0 Thie agrees with the result found in the previous Art. Thus M = ( is the neoessary and B'Ujficient condition that the parametric curve,

form a conjugate system We have seen that when the lines of curvature are taken e. pare.metric curves, both F = 0 and M = 0 are satisfied. Thus th linSB of CU,rvature form an orthogonal conjugate syGtem. And the: a.re the only orthogonal conjugate system. For, if such a. system o curves ensui, and we take them for parametric curves, then F = I and M = 0. But this shows that the pare.metric curves a.re the, lines of curvature. Hence the theorem. Ex. 1. The parametrio ourves a.re CODJUgate on the following surf&oea. (i) a eurf'aoe o{ revolution .1D-=u.cosrp, y=u&nc/,, 1=/('IJ,)i (tl) the surface genere.ted by the tangents to a ourve, on which

R=r+-ut, (u,, p&r&meters); (ili) the surface IC=

b> c, the ollipeotd ;:

+·r~ +

~=l hn.a umb1hci a.t the points

Y... o, z'-~,~~-:..b') al-& I


al-rfd •

O. Tho only dcvelnpn.bln surfao01:1 which have 1eomotrio Imes of ourvn.1lLl'O 01thor comoa.l or cylmdr1c1~l.

I. Taking tho &"lymptotio hncH ns pN•n.mctr10 curves, a.nd evalun.ting n', r'] along tho d1roct1011e vcaconHt. o.nd U=COil8t. 1 vonfy the values '-::-JL for the tonnonH of tho Mymptotio lmCR !Ji. Show that the mcir1dm11H n.nd pru-u.llol1:1 on 11, sphore form an isometno .em, o.nd dotcrmino thu 1aumotric p 1r1uuetors. , 3. Fmd tho t1Bymptot10 lmOH on tho ilnrfR.Ce :,;s:1.a(l+cosu)ootv, y-==-a(l+coRu)1


4:. Provo tlrn.t tho prudnot of the n,du of normo.l curvature

10 ooojuga.te 0t10n11 1s n miuitnum for lines of curvntnro. 5. A cnrvo, winch tonchos A.II Mymptutio Imo 11,t P, a.nd whoso osoulo.tmg 1c is not tiLngmtrn.J. to tho 1mrfucu at 1'1 has l' for a point of inflaot1on. 8. The normn.l curvnture 111 n. dircctiou porpeud1oulo.r to a11 asymptotic JB twico the me,m nurnml cnrv1~t11ro , 7. Show tho.t tba umbilici of tho Hurfo.ce

a. s11here. B. E1a.niine tho curvn.turo. a.nd J\nd the linos or aurva.tnra, on the ilco zy,-abc.


9, Show th,t the curv,turo of an Mymptotlo lino, aa given . 38, ru,y be oxpr...,ud


(26) of

(r1 • r r:1• r'' - ra • r' r1• r'')/ 11. 10. Tho fLMymptut10 linoM nn tbo huhcoi..F} F,-½E,=!F+~G' Solvmg these for I and~ we have

l=+(GE,-2FF,+FE,)j ..................... (2). = 2H' (2EF, - EE, - FE,)


Agam since r, • r,.=½E, and r,• r,. = ½G., we find from the second of (1), on forming the sce.lar product of each mde with r, and r, aucceeS1vely,

iE,=mE+p,F} jG,=mF+i,G' Solving these for m and p, we have


m= 2Ii'(GE,-FG,) 1 p. = ~ (EG, -FE,) , .................. " .. (S). 2



,rly, usmg the reJ.,tions r, • r,. = F, i from the tlurd of (1)


n= ~~I• (2GF,-GG, -FG,) 1 v = fili• um. - 2FF, + Ila,>

nnd r,. r,. = ½G,,


............... (4),

rmnlnc (1), with the v,ilucs of the coofficient.s• given by (2), :I (4), aro the ~qmvnlcnt of Gr,uss's fu,~111,lue for r11 , r,., r,., a.y bo rcforrutl to unclcr thia ntimo. en the ptirnmotrie curve• nrQ ortlw,qm,a!, the vnlucs of the coefficients e.ro grc11tly Hinzplilicd. For, in tins cnso, F= 0 •= EG, ao thnt E1 N~ ' r11 = Ln + "il!J r1 - 20 ra r,. = ,.'f n + 2II'. 1> r, + 2G,(,1 r, -N



n - 21~, r 1 +

1 • • ... • •••• .... •


a, 2(} r:1

, nrn unit vt>OLOl'H p11mll"I to r, nncl r,, we h11vo




r, =-;;u·

a., b, n fi,rm n ri,L:ht,-lum,l1•il HyHt,nn uf' unit VL>ctorH, mnt,111illy 1Jic11l,1r. l!'i,1111 l,J,...,, f11r11111l111! wo ch•cluc,• immuclilltcly thnt ila. /, !I', •

~It=".,/ g


n - '}./1 b

f:l, ;Ju=-;_;gn+2ifb





'ih, -

"..;u n + 211 a





i>u "JU n - 2lf a

erivntivca of a nrc perp1•n..r., = M1 n+'111or, + µ.,r,+ Mn, +mr,, +µr,.. If in this we substitute a.gain from (1) the ve.lues of the second derivatives of r, and also for n, and n, from .A.rt. 27, we obtain a vector identity, expressed in terms of the non-cople.ne.r vectors D, r,. r,. We me.y then equate coefficients of like vectore on the two sides, and obtain three scalar oquations. By eque.tlilg coefficients of D, for example, we have

L, + lM +"J,,,N = M, +mL +µ.M, that ie

L,-M,=mL-(!-µ)M -..N........... ... (7).

Simile.rly from the identity ;; r,. = ~ r,., on substituting from

(1) the values of r,. and r,. we obtain the relat10n M,11 + m,r, + ,,_,r, +Mn,+ mr,, + ,..r,.

= N1D +nir1+v1r:11+Nn1 +nru +vrll. Substituting age.in for the second derivatives of r and for n,. n, lil terms of D, r,. r., and equating coefficients of n on the two aides of the identity, we obtain that is

M,+mM + µN =N, +nL +vM, M,-N,=nL-(m-v)M-µN .. ........... (8).

48, 44]



The formulae (7) e.nd (8) are frequently called the Oodaezi equations. But as Ma.m.e.rdi gave Blirula.r results twelve yee.ra earlier than Codazzi, they a.re more justly termed the Mainardi-Oodazri relations. Four other formulae are obtained by equatmg coefficients of r, and of r, in the two identities: but they a.re not independent. They are e.11 dedumble from (7) and (8) with the e.id of the Gauss characteristic equation. 44. Alternative e:r:preeelon. The above relations may be expressed in a different form, wh,ch is sometimes more useful By differentiating the relation H• = EG- F• with respect to the parameters, it is easy to verify that H,=H(l+µ,), and H, =H(m+ v).

~ (~) =


¥;- ;, H,

=~-;(!+µ,), e.nd similarly

f.(;)=~- ff,n, =~-;(m+v),


o(M) -a,,:o(N) 1 M N 71 =Ii(M,-N,)-H(m+v)+ll(l+µ,)

aii It

= (nL-2mM + !N)/H.................. (9), in virtue of (8). Similarly it may be proved that

k(:)-f. (i) = (vL-

2µ,M HN)/H ...... (10).

The equations (9) and (10) are au e.lternative form of the Me.me.rdiCodazzi relations. We have seen that if six functions E, F, (}, L, M, N constitute the fun damen ta! magm tudes of a surface, they are connected by the thiee differentia.l equations called the Gauss cha.ra.ctenatic equation and the Ma.me.rdi-Codazzi relations Conversely Bonnet has proved the theorem: When Bia, fundamental magnitud83 are gwen, satisf'll'ng the Gaus, charact,,,....t;,,c equation and the Mairw,rdi--




Codazn ,-elations, they dstermine a surface uniqusly, emespt as to position and =entation in spacet. The proof of the theorem is beyond the scope of thia book, and we shall not have occaa1on to use 1t, •45, Derivatives of the angle .,, The coefficiente occurring in Gauss's formulae of Art 41 may be used to express the derivatives of the angle ., between the parametric curves. On d1fferentiatmg the relation


tano>=F with respect to u, we have eeCSMei,1 =

FH,-HF, Jj'•

Then on eubst1tutmg the value sec'o>=EG/F•, and multiplying both sides by 2HF•, we find

2EGH .,, = F(2HH,)- 2F,H• =Fi(EG-F•)-2F,(EG-F•)


= F (E, G + EG,)- 2F,EG

=Hence the formula

2H• ("/..G + mE) .

.,,=-H(1+;) ................(11)

And in a similar manner 1t may be shown that

,.,=-H(i,+;) ....



1. Show th&t the other four relations, ermils.r to the Ma1nard,-Cod..., relations, obt&ulable by equatmg coeillcionta of Art. 43, are eqwvalent to

r, and


r, m

the proof of

FK-,n,-Z,+m~-••• FK=,;-v 1 +mµ.-'M., EK-A 9 -1'1 +lp.-m>..+Xv-µ.1,

GK=n 1 -fflt+ln-m2 +,,w-np..

!2. Prove that these formulae m•y be deduced from the Ge.UJ!S characteristio equation and the Mainardi-Codazzi relat,one.


Fora,th1 Diferential Gromd'l', p 60.





Provo the relations

~ (~)-~ (%)-BK, (B") -s.a (B"') o= BK.,

ilua 0 ,ing the formulae in EL 1

4, If " •• the a.nglo between the parametnc cnrvea, p,ovo that

-..,,-~(~)+~(¥;)+BK -~ (1;;")+~(~)-BK. 5, If the aaymptct,c ho'"' are taken aa parametric ourv.., abcw that the .&1nard1~Coda.zz1 rela.t1ona beoome ~=!-,.,


ODCB deduca that (0£ Art, 44)

111-~+~. 11,.=~-~. 2m•!Ji-~, i•=t+,. 8, When the parametr10 curves a.re null hom, abcw that the lllamardJ.. odazz, relattona may be upreaaed




y-a:;;lcg:,, M-aw1cgF'

1CI the GaUIIB cbaracter1Stic equ&tton aa

LN-Jf'=li'.,-F,:,. 7, Whon the hnaar element ia of the form

d,I.; (dv'+do'), 10 Ma111&rdi-Oodazz, rolatiODS are

L,,-.ll',=½t,. = 0, e.nd the curves u = const. provided n = 0 When the parametric curves are orthogonal, these conditions are E,= 0 e.nd G, = 0; so that the curves v = const. will be geodesics If E is a function of u only; and the curves u = const. If G JS a funct10n of v only. Another formula for the geodesic curvature of " curve m&y be found in terms of the arc-rate of mcrease of its inchn&tJ.on to the pe.mme~nc curves Let (J be the 'inclmation of the curve to the





parametric curve v - const, measured m the positive sense. Then since, by Art. 24 and Note I E,,: + Fv' = ./E cos 6. we have on differentiation

a, (EU' +•·• Ja.A d./E COB (J ds 1=,ls

·'E- Blll . (Jd(J a&


= 2~(E,u' +E,v')(Eu' + F,l)- Hv· ~. Now, if the curve ie a geodesic, the first member of th1S equation ie equal to ½(E,u'•+ 2F,u'v' + G,v"). On snbetitution of this value we find for a geodesic


H•>.. ,


H'di=- 1r" -7rv'. Thus the rate of increase of the inclination of a geodesic to the para.metnc curve v - const. IS given by d(J H( ,._.,,, +p,v. ·') 'di--E Now the geodesic curvature of a curve O ie tangential to the surface, and its magnitude 18 the arc-rate of deviatwn of O from it.a geodesic tangent. Thie ie equal to the difference of the values of rl,(J/da for the curve and for its geodesic tangent But its value for the geodesic has just been found. Hence, if d(J/da denotes ibs value for the curve 0, the geodesic curvature of O IS given by


+:{71.,,: +p,v') .................(21).

Or, if~ ie tbe inchnat10n of the parametnc curve 11, = conet. to the curve (Fig. 11, Art. 24), we may wnte this d(J >..ifo: p, . '"•='di+ "'"]f'" Bin~+ ,/E Bin (J ............(22).


In the particular case when the parametnc curves a.re orthogonal, em~= cos 6. .Also the coefficient of em~ becomes equal to the geodesic curvature of the curve v = conet., and the coefficient of em (J to that of the curve u = const Denotmg these by ...,. and ,.,. respectively, we have Liouvi.lWsformvl,a, '"•=:+,.,.cos (J +,.,.sin 6 ............... (23).



•&&. Ezamples. (1) BonMt', 7armula for the geodeeio curv&t1ll'O of the curve (u,

By d!fl'erentiat,on we h•ve

,t,1 u'+tf,,v=O ,. ...... .. u'



........... (,



eo tb&t

where 8=.JE, c/>, + Ee/>,')/ H• must be a function of c/> only, OT a oonstant. The condition is also s1,jfioient. For d,s'-

ed,c/>,)dudv + (G- e,f,.') dv'

and this, regarded "'1 a funct10n of du and dv, 1J3 a perfect square, m virtue of(«) bemg satisfied. We can therefore wnte 1t as Ii'd,Jr', so that

da' = sdc/>' + D'd,[,',

59, 60]



proving the sufficiency of the condition. In order that ,f, may be the length of the geodesics measured from ,f, = 0, it 10 necessary and sufficient that e= 1, that 10 G,t,,• - 2F,f,, ,f,, + E,f,,' = H• ....•.......... (30'). 60. Geode■lo ellipses a.nd hyperbola.a. Let two independent systems of geodesic parallels be taken as parametric curves, and let the parametric varmbles he chosen so that u and v are the actual geodesic distances of the p01nt ('u, v) from the particular p

Flg lB

curves u = 0 and v = 0 (or from the poles in case the parallels are geodesic circles). Then by Axt. 59, since the curves u = const. and v = const. are geodesic parallels for which • = 1, we have E=G=H•. Hence, if ., is the angle between the pare.metric ourves, it follows that E= G =___]_, • F- cos.,


BlD a>

so that the square of the linear element 1s du• + 2 cos ., dudv + d'lf' rls' = am•., ........•...... (31) . .And, conversely, when the linear element is of this form, the pare.metric curves are systems of geodesic parallels. With tlus choice of parameters the locus of a point for which u + v = const. is called a geoduio ollip••· Similarly the locus of a point for which u-v = const. is a geoduio hyperbola. If we put u=l(u+v), ii=l(u-v) ............. (32), the above expreSB1on for ds' becomes

dB' = .±!.:.. + diP s1n1



................. .(83),




showing that the curves 11, = canst. e.nd ii= canst. a.re orthogonal. But these a.re geodesic ellipses and hyperbolas. Hence a sylltem of

geodesio eUipses and the corresponding system of geod..-io hyperbolas (/1['8 orthogonal Conversely, whenever ds' is of the form (33), the substitution (32) reduces it to the form (31), showing that the parametric curves in (33) a.re geodesic ellipses and hyperbolaa. Further, if 8 lB the inclinat10n of the curve ii= canst. to the curve • = const., it follows from Art. 24 that cos 6=coe


Bin O=smi,

e.nd therefore Thus the geodekio ellipses and hyperbola,; bi.seat the angles beflween

the corresponding system11 of geodesic p(l/f'alldls. 61. Llouville surfaces, Surfoces for which the linear element is reducible to the form d.s'=(U+ V)(Pdu•+Qdv') . ............ (34), in which U, P are functions of u alone, e.nd V, Q a.re functions of v alone, were first studied by Liouville, and a.re called e.fter him. The pare.metnc curves clearly constitute e.n isometric system (.Arl. 39). It is also easy to ahow that they are e. system of geodesic ellipses and hyperbolas. For if we change the parametric variables by the eubet1tut1on dii .17' av

,rPdu= ,/'f]'

-vQdv= vV'

the parametric curves e.re unaltered, e.nd the linear element takes the form

M=(U But this


+V)(~' +~).

of the form (33), where ,w


1 0,_


sm2=u+v· cos2-u+v· Hence the p(l/f'a1t1etric cwrves (/1['8 geodesic ell,pses and hyperbolas. Liouv,lle also showed that, when M has the form (34), a. first integral of the diff'erent1al equat10n of geodee1ce is given by U am• 8 - V cos• 8 = canst ............. (36),




where Ois the inchnat1on of the geodemc to the perametnc curve v = const. To prove th!S we observe that F = 0, while

E=(U+V)P, so that E,= U,P+(U + V)P,,


G=(U+V)Q, G,= U,Q, G,=V,Q+(U+V)Q,.

Te.king the general eque.t10ns (4) of geodee1cs, multiplying the first by - 2u' V, the second by 2v' U and adding, we may e.rre.nge the reeult Ill the form

-:a(UGv'•- VEu'')=u'•v' {(U + V)E,- V,E) -u'v'• {(U + V) G,-U,0). Now the second member vanishes 1dent10e.lly in virtue of the preceding relations. Hence UGv'• - VEu'• = conat., which, by .Art. 24, 1a equivalent to U am• 0 - V cos' 0 = const. e.s required. EXAMPLEA VIII l. From formula (21) deduoe the geodesic ourvatura of the ourvea ,-oonst.. a.nd u,....conat. !2. When the ourves of &D orthogonal system h&ve oonst&nt geodamo ourv&ture, the syet.em 1s 1sometno. 3. If the ourvee of one famtly of a.n 1sometnc eyetem ha.ve constant geodesic ourva.ture, ao a.Ula ha.ve the ourvee of the other family. 4. Straight hnes on a surfe.oe a.re the only ••ymptot10 !moo whioh e.re geodesics. 6. Ftnd the geodea,oa of e.n elbpao1d of revolution. 8. If two f&milies of geod.681cs out at e. constant angle, the surfa.oe 1B developable. 7. A curve 1s drawn on & oone, semi-verbe&l angle a, so ea to cut the generators at a. oonsta.nt a.ngle fj. Prove that the torsion of its geodemo tangent 10 inn{Jcos{J/(Rte.na), whore R 18 the diste.noe from the verteL 8. Prove tha.t any ourve is a. geodwo on the surface generated by its b1normela, and a.n asymptot10 hne on the surface generated by its prinoipal norme.l.s 9. Fmd the geodesics on the ce.teno1d of revolution

u-o ooah~.




1 O. H a geodesio

on a. eurfe.ce of revolution outs the mencha.ne at a oon~ i!t&Dt 011gle, tho surface IS • right oylmder

11 . If the pnnc1pa.I. norm&ls of a. curve mt!!1'8eot. a. :fixed line, the curve is geod0S10 on a. surface of revolut1on1 s.nd the :fixed hne 1B the ax::1.s of the surface. 1,g. A curve for wh1oh tc./r is constant ie a. geodesic on & oyllnder, e.nd a


oorve for which


(T/1e) 18 constant 1B a geodes10 on e. cone.

l 3, Show that the family of cunos given by the difl'erentis.! equ&tion Pclu+Qdv-0 will oonstitute & system of geodOB10 para.Ilels provided



HQ ) ./E(l'-21!'PQ+Gl~



HP \ ;/EQ'-21!'1'Q+G]it}'

14. If, on the geodm.N through a. point 0 1 'Point, be taken at equal gBD~AIJ dutanC1B1 from 0 1 the lot1UB of the pcnnt, i, an ortlwgon,al trajeet... At points common to the two surfaces (1) a.nd (2) we have

fOOs~+~~+~~+~-Zal'~+~~+~-~ We may regard this a.a a.n equation for determmmg the values of X correspondmg to the confocals which pa.ea through a given point {01, y, e) on the surface (1) It is a cubic equation, one root of which is obviously zero. Let the other two roots be denoted by u, tJ. Then, because the coefficient of x• 1B unity, f (X} is identioally eq=l to the product X (A- u) (X - tJ); tha.t is

X(A-aj(A-~s~+~~+~~+~-Zal'~+~~+~ If in this identity we give X the values - a., - b, - c in succeaaion, we find af'- a.(a.+u)(a.+tJ)' - (a.-b)(a.-c) b(b+u)(b+v) 1/= (b-a.)(b-c} ..................... (3). e'=o (o+u)(o + v) (o-a.)(c-b). Thus the coordinates of a pomt on the qua.dnc (1) are expressible in terms of the pa.re.meters u, v of the two confocals pa.eamg through that point. We ta.ke these for parametric variables on the surface. It follows from (3) that, for given values of u a.nd v, there e.re eight pomta on the 1111l'fa.ce, one in ea.eh octent, symmetrically situated with respect to the coordinate pla.nea.

12, 63]



In the case of an ellipsoid, a, b, o are all pomtive. Hence ,t, (- o) a negative, ,f, (- b) positive, and ,f, (- 12) negative. Therefore, if u a greater the.n 11, we have -o>u>-b, -b>11>-a. l'he vo.Iuee of u e.nd v are thus negative, aud are separated by-b. For e.n l1yperbolo•d of one sheet o 1s negative, so that ,f,(ao) lB positive, ,f,(-c) negative, ,f,(-b) positive and ,t,(-12) negative Therefore u>-a, -b>11>-a. Consequently u is pomtive and 11 negative, the root between - c 11.nd - b being the zero root For e.n l1yperboloid of two sheets both b e.nd o are negative. Hence ,f,(ao) is pOS1tive, ,f, (- c) negative and ,f, (-b) positive, so that the non-zero roots a.re both pos1t1ve and such that u>-c, -c >11 >-b. Thus both paro.meters are positive, and the values of u and 11 are separated by - c. In all cases one of the three surfaces through (ai, y, 11) 1s an ellipsoid, one an hyperboloid of one sheet, and one an hyperboloid of two sheets. Any parametric curve 11 = const. on the quadric (1) is the curve of intersection of the surface with the confocal of parameter equal to this constant v. S1m1larly any curve u = const is the hne of intersection of the surface with the confooal of parameter equal to this constant u.

ea. Fundamental magnitudea. If r is the distance of the point (a,, y, 11) from the centre of the quadric, end p the length of the central perpendicular on the tangent plane at (m, 'JI, 11), we have and

r" =""+'JI' +.o" =(a+ b+o) +(u + v)} .!.-~+i+~=~ · ....... (4,),

p' a• b' rf' abc Also on calculating the partial derivatives a:,,

E = "1° + '//,' + •1'=

ai,, etc., we find

i"°(a+u)~:~:~ (o+u}

F-a:,o:.+'JJ,'/1•+•,,.-o I



0-.,, +y, +s.

11(11-U) •4(12+11)(b+11)(0+11





The norme.l has the direction of the vector (~, the eque.re of th1S vector D -






!,D,and Bl.Ilce

eque.l to 1/p', the urut normal is

1!!) C

(a+u)(a+v) (✓be = uv(a-b)(a-c)'


uii (b-c)(b- a)'


The second order magnitudes are therefore

L-n•r -

-!Jabc (u-v) uv (a+u)(b+u)(c+u)





ab (c + ,i)(c + uv(c-a)(c-b)'

'l ..... (6).


(v-u) =n•r,.=4 uv (a+v)(b+v)(o+v),' Smee then F = 0 and M = 0 the parametric ourves are lines of curvature. That is to eay, the lines of curvature on a. central qua.dnc are the curves in which it le cut by the confoce.le of different epe01ee. The prrnmpal curvatures are then given by

"•=~= E !"

Jiihc) UV

"•=~=!Jabc ... · .......... ,(7). G


Thue, along a line of curvature, the principal, curvature vaM as the cube of the other principal curvaliure. The first curvature is

J=i(u)..... . ..


103, 104]



Then any displacement (da:, dy, dz) on the.t surface 1s such tha.t



+~cf,'(u)] du=Oi

~da:+~dy+~d-+[*+~cf,'(u)]du~o • But at the foci


g, g, and therefore for any chreot10n In the tangent ple.ne at a focus we must have

~da:+ oj_ dy+~ dz=}.. (~ d,o+ ~ dy+ ~dz). • ¼ b • ~ ~

Thus the norme.1 to the surface e.t the focus is parallel to the vector

(~->,.t ~-}.,~, ~->..t) .........(31), which is independent of the assumed rele.t1on between the pare.meters, and is therefore the so.me for all surfaces of the congruence through the given curve. Hence ell these surfaces touch one another at the foci of the curve. Again, the equation of the focal surface is the elimmant of u, w from the equat10ns (32) and (34). At any pomt of the focal snrfu.ce we have from the first of these

!da:+idy+~a.z+!~+r•-oi ¼ ¼ ~ ~ ¼ ' a,;dm+ ijdy+ az ds+a,;, du+a,; dw= 0 and therefore, m

virtue of (34 ),




aa, da:+ ay dy + az dz =-b. =b.= -x. ~da:+~dy+~dz oy ""




Thus (dm, dy, dz) is perpendicule.r to the vector (31), which is therefore norms.I to the foes.I surface. Hence at a focus of the curve the fooal surface ha.s the so.me tangent plane as any surface of the congruence which passes through the curve. The theorem is thus established. It follows that any 11Urjacs of the congruence touches



the focal surface at the Jo~ of all •ta curves The tangent plane the focal surface at the fom of a curve a.re called the focal plan, the curve. 105. Normal congruence. .A. curvilmear congruence is, to be normal when it IS capable of orthogonal mtersection b fanuly of surfaces Let the congruence be given by the equat1 (31). .A.long a particular curve the parametera "• v are conall and therefore, for a displacement along the curve, we have differentiation




g) = J(;;a, g) = Jff, g). y, z \.m, y



If ID thlS equation we substitute the values of "• v m term, a:, y, z a.e given by (31), we obtam the differential equation of curves of the congruence in the form da; dy dz (381

:x=-y=z ....... ............

where X, Y, Z are independent of "• v. If then the congruenc, normal to a surface, the differential equation of the surface must Xda,+Ydy+Zdz=O ... ...... (39) In general this equation 1a not mtegrable. It is well known fr the theory of differential equations that the cond11Ion of integ b1hty IS


("y az oZ) ay + Y(oZ a,. _ oX) az + z (oX ay __ oY)-o. a.,,

If this condition is oatisfied there is a family of surfaces satisfy the equation (39), and therefore cuttmg the congruence ort gonally. Ex. l. The oongruenoo of circles l't"+my+m:=u, xl+y1 +r=11 ball for 1te differential equa.t1on d:,,

dy d., 11,y-1iU- ls-nz """'m:.,;-ly" Renee they are norme.l to the surfaoes given by (11,y-m,) d,: +(ls-=) dy+ (mm-!y) d.,=0. The oond.ition of mtegrab1hty 18 sa.tiefied, e.nd the mtegra.1 may be &xp1'61!





where (l is an a.rbitre.ry oonsta.nt. Th.te represents a fa.mily of pla.nes with common lme of intersection :r:/l=y/1a=1/n.

Ex. 9:. The congruence of conios y-•=u, (!l+•l'-4.,,=v has a. differential eqWLt1on .!!!.._=,fy=da. y+, It IB normal to the surf&oe, givau by (!I+.) ck+dy+d.e=O. The oondit1on of mtegrab1hty IB 88tIBfied, and tbe mtegra.1 IB

y+•=ce-•. E:z::.


Show th&t the congruence of c1roles ~+y1+z'=U'lJ=-VII has the d1fferentieJ eque.t1on

~--~-~ 2xy-2:r:,'

lfl-y1 -e'

and 1s out orthogona.lly by the family of spheres

:r+y'+11~-a:11. EXAMPLES XIII

ReotiliMar Ormgn.umcu 1. The current point on the m1ddle surface 1B

R=r+td, where


The condition tha.t the sur.faoe or reference m11.y be the middle surfaoe is

eo+ga-f(b+b') 53. Prove that, on ea.oh sheet of the face.I surfaoe1 the ourvea correepond.mg

to tb!3 two fa.mW.ea of developa.ble surf&088 of the congruence &re conJug&te. a. The ta.ngent ple.nee to two oonfoca.l que.dnos at the po1nte of oonta.ct of a common tangent a.re perpendicular. Hence show tho.t the common ta.n gente to two confocal quadncs form a. norme.1 congruenoe.

4. If two surfa.oes of a. congruenoe through a. given ray &re represented 011 the unit sphere by ourves which out orthogonally, thell' lines of str1obon mee1 the ra.y a.t pomt.B equ1d1Ste.nt from the middle point. 6, Through ee.ch point of the pla.ne •=0 • ra.y (!, m, n) 10 drawn, euoh lh&I !=ky, ,,._ -•.., "= ✓1-l:'(.,,,+y'). Show that the oongruenoe so formed 18 1eotrop1a, with the pla.ne .1-0 ea m1ddlo surface.




6. In Ex. l, Art 102, prove conversely tb&t, if the surfa.ces a.re •pphcable, the congruenoe is 1Sotrop1c. 7. If (l, m, n) 1s the unit normal to & m1mm&l surface e.t the current point (111,11, ,), the htte P"'allel to (m, -!, n) through the pomt (",.l'• 0) generates a norms.I congruence. 8. The hnea of etnotaon of the mea.n ruled surfaces he on the middle surface.

9. For any obo1oe of pa.re.meters the d:dferentiaJ equation of the mean surfaces of a. oongruence IB a,g-(b+b')f+c, "'d"'+(b+b')dudv+odv' 2h' ed•'+2fdudv+gdv' ' 1 O. The mean pa,ra.meter of dlBtribution (Ai;t. 99) of a. congruence ie the squ.&re root of the difference of the squares of the dISta.noea between the hmits and between the fooL I 1. If the two ebeete of the foe&l surface 1ntereeot1 the curve of mteraeotJ.on m the envelope of the edges of regremnon of the two families of developa.ble eurf'&o8B of the congruenoe.

l ll, In the oongruence of etrrugbt httes wb1oh mtereeot two twietod curves. whose arc-lengths are,, ,'1 the d&erentlR.l equ&taon of the developable surfaoes of the congruence is d, di= O The foca.l planes for a ray a.re the pl&D.es through the rs.y aod the t&ngeuts to the curves at the points where it cul-.s them. 13. One end of an 1nerlemnble thread is attached to a fixed pomt on a smooth surf~ and the thread ia pulled tJghtly over the surface. Show that the poes1ble positions of its str&ight portions form a norms.I congruence, and that a pa.rtiole of the thread desCI'lbee a. norm&! surface.

14. In the con9'f"U811,oa of latn,gmt, to 01UI ,y,unn of ruymptotUJ lina, on a 9wffl l'Urfaca, S, show that the two sheets of the focal surface oomc1de w1th ea.oh other and Wlth the surface 8, and that the distance between the hDUt pomts of • ray 18 equal to 1/.r=Jf, K bemg the epeoifio curvature of the surface 8 at the point of contact of the ray Take tbe eurfa.ce S as director eurfaoe, the given eyetem of a.eymptot10 Imes 88 the para.metric curves ii--oonst., a.nd thell' orthogonal traJeotones as the curves u = canst. Then, for the surface S, L=O, F=O, 80






•om these 1t is eamly venfred th&t

a .... o, b-

E, -"i:JE'

b' 0

... ,


c- 2JE'

•-:Jo· f=-f~, o=4M'!;o,•,

A•-~t11e eque.ti.on (15), for the d.i.eta.nces of tb.e foci from the director surface, duoes to p2-=0 -Thus the f001 001no1de; and tbe two sheets o! the fooa.l rface oommde with the surfd.08 S The congruence 1S therefore para.bolio .rl. 99). 811ml&rly the equation (7), for the diste.ncee of the lmuts, reduoes to

r=f:.=~, l


oue the d1&tanoe between the hm1ts 1s 1/.J-=ll. 16. When the two sheets of the fooal surfaoe of a reotilinea.r congruence 11n01de, the epeo1fio curvature of the fooal ear.face at the point of oonta.ot of

ray 18 -!/!", where! 18 the dietence between the hlnlts of the r&y.

18. If, in a normol oongruenoe, the dist&noe between the foci of a ray ia LB ee.me for all rays, show tb&t the two sheets of the fooal aurfaoa have their ,ecllio ou.rva.ture oonsta.nt &Dd negative. 1 7. Raye are 1nc1dent upon a refleotmg surface, and the developablee of 1e 1noident oongruenoe are reflected into the develop&bles of the refleoted

,ngruenoe Show tlmt they out the re6.eotmg surface 1n CODJugate hnee 1 8. When e. congruence conS1sts of the tangents to one sytstem of lines ' ou?'V&tUl'e on a surfaoe, the focal chstanoes &re equal to the radii of geodesic irvature of the other system of Imes of ourv&ture.

18, A neceaR&ry and euffi01ont condition that the tangents to • f&mily of irvea on a sur.f'a.ce m&y form a normal oongruenoa l8 thAt the ourves be


!30, The enre1DJl!ee of • stra1ght hne, whose length ia ooneto.nt &nd h088 dll'80taon depends upon two pare.motara, a.re ma.de to deecnbe two 1rl'aces e.pphca.ble to each other Show th&t the positions of the luie form o. isotrop10 oongruence. !31. The epbenO&! reproeent&l!.on& (Art 94) of the dovelopable surfaces

r an iaotropio oongruence are null lmea. !J!J. In an isotropic oongruenoe the envelope of the plane whloh outs ray orthogom.Jly at its 1DJddle pomt 18 &mmim&l eurfaoe.




Cu,,nlz11.8ar Oongru,,mca& 138. Prove that the congruence ,:-y=u (1-:e) } (,:-y)'(:e+y+•)=• IS normal, being out orthogon&lly by the f&m1ly of surfaces ys+""+"!i=o'. 134. Sbow that 2.1:'-y'-u', 3y'+2t'=•'

represents & norme.l oongruence, cut orthogone.lly by the surfaces "11'-oa' !35. Four surfacee of the congruence pass through a. given ourve of the congruenoe Show th&t the cross-ratio of thm four tangent plaD.es at e. point of the curve lS independent of the point ohosen g 8. If the ourves of & congruence cut & fixed ourve, O, ea.oh pomt of mtersect:J.on IS & foo&l pomt, uoleu the ta.ngents a.t this pomt to eJl curves of the congruenoo wbicb p&BB through it, are ooplenar with the tangent to the curve O &t the same pomt.

g 7. If aJ.l the ourves of a. oongruence meet o. fixed curve, this fixed curve lies on the focal surface. 138. Show that the congruence


arj, (z)+b,f,(/l)+•x rj,(:e)+,f,(/l)+x(s)=• norm.a.I to & f&mlly of surfaces, and determme the family. 139. Fmd the pere.llel plane oeotions of the surfaces rj,(z)+,f,(/l)+x(•)=u whioh comtitute a. normal congruence• and determine the family of surfa.oes which out them orthogon&l.ly


80. If a. congruence of Oll'olea is out orthogone.JJ.y by more than two surfe.cea, 1t 1S cut orthogonally by & famtly of eurfaoes Such a congruoooe 1s oaJled a ayclio ,y,tem Non. The author h&a recently shown that ourvilinea.r oongruenoes may ho more effeoti.vely treated along the same illlee as reottlmea.r congruenoae. The uistence of a. hm.it l!llurfa.ce and a surface of etnction is thus easily esta.blished, end the equet1ono of these surfaces ere reedtly found. See Art. 129 below.

CHAPTER XI TRIPLY ORTHOGONAL SYSTEMS OF SURFACES 106, A triply orthogonal ayatem coDS1Sts of three families )f surfs.cea u(m, y, .r)=conat.t 11(m, y, .r)=const. .. ................ (1) w(m, y, •)=const which are such that, through ea.eh point of apace paaaes one and )nly one member of each family, each of the three surfaces cutting the other two orthogonally. The mmplest example of such a system IS afforded by the three families of planes m=canst, '!/=canst., • = const., paro.llel to the rectangular coordinate planes. Or again, if apace 1s mapped out m terms of spherice.! polar coordmatea r, 8, rf,, the ,urfaces r = conat. are concentric spheres, the surfaces 8 = const 1.re coaxial Cll'CU!ar cones, and the surfaces rf, = canst. are the meridian planes. These three fam1hes form a triply orthogonal ,ystem. Another e:rample is afforded by a famtly of pe.rallel surand the two fam1hea of developables in the congruence of ,ormala (Arts. 74, 100). The developables are formed by the 1ormala along the Imes of curvature on any one of the parallel 1urfaces. As a last example may be mentioned the three families ,f quadr1ce confoca.l with the central qna.dnc


~+i+~=l. a



t is well known that one of these is a family of ellipsoids, one , family of hyperbolmde of one sheet, and the third a family of 1yperboloids of two sheets Thie example will soon be considered n further detail

107. Norma.la. The values of u, 11, w for the three surfaces hrough a pomt are called the ou,.,,.linear ooordinat"8 of the pomt. 3y means of the equations (l} the rectangular coordmates m, y, •• nd therefore the position vector r, of o.ny point in apace may be xpressed in terms of the curvihnea.r coordinates. We aaaume that



this he.a been done, e.nd we denote partial derivatavee with res, to u, •• w by the suffixes l, 2, 3 respectively. Thus i)r or o'r

r1=az;, r,=a:;;, ru=ouov'

e.nd so on. The normal to the surface u = const. at the pomt (a:, g, e parallel to the vector (au OU

'°--;, aii .

asau) .

Let a denote the ,init normal m the direction of u increas1 Similarly let b e.nd c denote um~ normals to the surfaces co





./ a



l'ig ~•.

and w= conet. respectively, in the directions of • increasing a w mcreasmg. Further we may take tho three families ia tt cyclic order for which a, b, o are a right-handed system of w vectora Then srnoe they are mutue.lly perpendicular we have a,b=b•o=0•a=0 } and a=b X o, b=0 X a, 0=8 xb ............ (2)And, because they are unit vectore, a'=b'=C'=l ..................... (3). Since the norme.l to the surface u = oonst. JS to.ngentae.1 to t surfaces • - const. and w = const through the pomt cons1dere for a displacement ds in the direction of a both • and w a constant. Jn terms of the change du in the other parameter let


107, 108]



Thus pdu 1B the length of an element of a.re normal to the surface u= const. The umt normal in this d1rect10n is therefore given by dr Iar 1

&=ds = pai. = pr.,

so that r,=pa (4), and therefore r,• = p'. S1m1larly if the elements of arc normal to the other two surfaces, in the directions of band c, are qd" and rdw respectively, we he.ve r,=qb, r,=rc .................. (5), and consequently r,' = q', r,• = r'. Thus r., r,, r, e.re a nght-handed set of mutually perpendicular vectors, so that r1 • ra = r1 • r, = r8 • r1 = 0 ........ , .. . . .. (6)

Further, in virtue of (2), (4) and (5), qr , r1xr1=pr1

r. x r1 =:Er~ ► q

r1 x r11

... (7),

=I!; r,

[r,, r,, r,]=pqr[a, b, o]=pqr ........... (8).


108. Fundamental magnitudes. A surface u = const. lB cut by those of the other two famihes in two families of curves, V=const. and w=const Thus for pomte on a surface u=const. we may take v, w as parametric vanablee S1mile.rly on a surface v = canst. the parameters are w, u and so on Thus the parametric curves on any surface a.re its curves of mtereect1on with members

of the other fam1hes On a surface u = conet the fundamental magnitudes of the first order a.re therefore E=r.'=q' } F=r11 •r,=0 ........... . 0=r,'=r'

. ........ (9),

eo that H• = q•1•, and similarly for the other surfaces Since F = 0 the parametnc curves on any surface constitute an orthogonal system. w.





To find the fundamental magmtudea of the second order we examine the second denvat1ves of r. By d1fferentiatmg the equat1ons (6) with respect tow, u, v respectively, we have

r,.,r,+r,,r,.=0} rm•r.+r11•r11=0 . ra1 • r1 +rs•r1e=O Subtracting the second and third of these, and comparing the result with the first, we see 1mmed1ately that r1 • rt8 = r11 • rn = Similarly r,, r,. = 0 · .... · "" · .... (lO),


Again, by d1fferent1atmg

r,• = p' with respect to u, v, w, we have


r 1 ,r11 .. (11), r1 •ra=P.JJ11 , ........ .. r1 •ru=pp e.nd therefore r1 • r 11 =-r1 • ru= -pp2} (12), r,• r11 =-r1 • ra=-pps with two similar sets of equat10ns. Now the umt normal to the surface 11, = conat. is r,/p, and the pe.iametere are •• w. Hence the second order magnitudes for that aurface have the values

L=.!.r,,r.,=-.!.qq, 1 p





..... (13).

N=.!r,,r,.=-.! rr, I p



Similar results may be written down for the surfaces •=canst_ and w = const. They are collected for reference in the table• Snrrace

u=const. 'Z7.,,,CODSt.


Para.met.ere ., 'ID

'°•" u,•


I b' > a'. The ooofocala are given by a,'J



a'H+b'+A+o'+A=l, for ddl'arent valuos of ). Hence the valuee of A !or the con!ooala through • given pomt ("', 1/, •) ..., given by the oub10 oqnat10n ,p().)a(a'+A)(b1 H)(o1 +l.)-'S,P(b1 H)(o'+).)=0 Lot v, ., 10 denote the roots o! tlwl equation. Thon, mnco the ooeffi01ont of ).I 11 equal to uo1ty, we have ().- u)().-•) ().-10)a(a'H) (b'H) (c'+A)-:z..1 (b'H) (o'+A) Ifm tlua identity we give Athe valttea -a.1, -61, _,jj 1n sucoesBJ.On, we fiod

..,_ca•+u)(a'+•l (a.1 +,o)l (a'-b')(a'-o') ll'=(b'+t m (JAl'/1 d,rsotion a,lO'fl!I the surface is th.s reao!VBd part of Vr/> in thru direction If c 18 the unit veotor in the direc1aon P.R, the rate of increase ofr/> m thie direction is therefore o • Vr/>- This may be called the dsnva;tive of r/> m the direction of c. If dr 18 the elementary vector PR we have dr=ods; and therefore the change dr/> in the function due to the displacement dr on the surface 18 given by d dr/>= dB ds-ds(o•Vr/>)

dr/>=dr•Vtf> .... •............. (2). From the definition of Vr/> it 1s clear that the curves r/>- const. will be parallele, provided the magnitude of V is the same for all points on the same curve; that 18 to eay, provided (Vr/>)' is a


• We lh&ll bon,ow the aolallon ond lermmology of lbree-par....,1r,o cldlennlul IDY&n&D.il.




function of rf, only Hence a necessary and sufficient condition that the ouries rf, = const. be parallels is that (V rf,)' is a function of rf, only The curves,[,-= const. will be orthogona.l trajectones of the curves rf, = const. if the gradients of the two functions are everywhere perpendicular. Hence the condition of orthogonality of the two systems of curves 1S • Vrf,•V,[,-=0. Although the gradient of rf, i,, mdependent of e.ny choice of parameters, it will be oonvement to have an expression for the function in terms of the selected ooordmates u, • This may be obtained as follows. If Su, So 1S an mfimtesimal displacement along the curve rf, (u, v) ~ const., rf,,.Su+,;,So=O. Hence a d1sple.cement du, dv orthogonal to thlfl is given by (Art 24)



do = Erp. - l!'rf,( The vector

V=(Grf,, -Frf,,) r, +(Erf,,-Frf,,)r, 1s therefore parallel to Vrf,. But the resolved part of thJB in the direction of r, is equal to 1 E F .j¾r, •V= (Grp,-Frf,,) .;E+ (Erp.- Frf,,) ./E

_H• - .~~ - .;Erf,,-H ./Eou' • Beltram.i's diBerentisl parameter of the 4rs1i order, .6.1 t/J, is the square of the map1tude of V (/J, that lS


HUI mued dtfferential pa.:ra.meter of the first order, A1 (tf,, \f'}, ia the soalar produot of the gradients of (/J a.nd

If; or

Darboux's ftm.alilon 8{~, ,f,)


A,(~, y,)=V~oVy,. the magnitude of the vaotor prodnot of V(/J &o.d Vtf,j

that u to aay The lnohnabon 6 of the onrve t=oonet


the onrve rf, =oouat. 1e alBO the mollno.tioa

of V\f, to Vt/J And, smce ooa1 B+Bl.D.1 8=f, 1t followafrom the last two equat10:ns1 on aquarmg and addlllg1 that A11 (~, y,)+8'(4>, y,)=(V~)'(Vy,) 1 =A,;,A,y,.




uch is H• times the denvat,ve of ,f, in the direction of r,. Hence ,d ,f, lB V/H•, or _(Gq,,_-F,f,,) (E,f,,-Fq,,_) V,f,H• r,+ H• r, ......... (3), is the reqmred 8"preSS1on for the gre.dient. We may regard this e.e the reeult obt&med by opemtmg on the 1ct10n ,f, with the vectorial differential operator 110h

V=j, r, ( G~ -F!) + :i, r, ( E~-Fiu)-~ ,at thie operator lB mve.rmnt ,s clear from the defimt1on of V ,f, nch lB mdependent of pe.rametere. The operator V plays a funde.mtel part m the followmg argument, for all our invariants e.rs prese1ble in terme of1t. Wben the paramefJMo OUrt/88 are orthogonal takee a e1mpler form. For then F = 0 and H• = EG, so that V

1 a 1 a =Er' ai; +ar, a,;•

form which Wlll frequently be employed when ,t nplify the calcule.tione.


des,red to

l!lz. Prove the following rel&tioos:

(V,t,)'-j;(E,t,~-2F,t,1,t,1 +flrf,,_1), v,i, • vt-fp[E,t,,+.-P(,t,,t,+,t,,t,)+O,t,,t,J, v,t,xvt=1(,t,,t,-,t,,t,)n. 115. Some applicat1ona, The gradients of the pe.re.metere ware given by

that d therefore





G (Vu)'= H•= G(V11 x VvJ',

Vu• V11---f't;=-Jl(Vux V11)1,


('vw'f=w=E(VuxVv)'. H then it is de811'ed to take as parametric V81'Jablee any two fnnotJ.ons ,t,. ,f,, the corresponding va.luee E, F, ll, Ii of the fundamental magnitudes are given by •l H•=(V,t,xV,f,)1 and E-li•(V,f,)', li'=-ll•V,t,•V,f,, 0=11'(V,t,)•.

H the po.re.metric curvee are rrrthogOMJ we h ..ve simply 1 1 (Vu)'=E• (Vv''f=0 In order that curves .,, - const. may be e. system of geodesic pe.rallels, E must be a function of u only (Art 56). Hence 11 MC6BS0.1'1J and 1111,jfi,,MM cond,non th.at th, ,t, = C011Bt. be geodmo parall.els u that (V 4')' be a fa.nation of 4' onLy. If the po.re.meter ,t, IS to measure the actual geodeeic dJStance from a fixed parallel, we must have (V ,t, )' =1 The following e.pphcatJ.on of the gradient will be required later.

If a IS a curve on the surface joinwg two points A, B, the definite integral from A ,to B of the reeol ved pa.rt of V ,t, tangential to the curve u

1: t•'v,t,ds-1~ dr•V,t,=

f d,t,=t/iB-t/iA,

t being the umt tangent to the curve. Thus, if A is fixed, the definite integral 18 a pomt-function determined by the pomtlon of B. H 4' is smgle-va.lued, and the definite mtegraJ IS taken roimd a closed clll'V8, 4'B becomes equa.1 to 4'• and the mtegral vanishes. When the path of mtegrat,on JS closed we denote the fact by a small Cll'C!e placed at the foot of the mtegral Sign. Thus

j V,t,•dr=O ...... 0

. ..........





Conversely, suppose that the vector I' is tangential to the surface,

e.nd that



1' • rJ.r ve.mshes for tnJery closed curve drawn on the

surmce. Then

J; I' • dr must be the same for all paths Joinmg A

e.nd B. H then .A is fixed, the value of the integral is a pomtfunctron tf, determined by the position of B. Hence, for any small displacement rJ.r of B, we have

l'•dr-dtf,=Vtf,•dr. This is trne for a.II values of dr tangential to the surface. Hence, aince I' and Vtf, a.re both tangential to the surface they must be equal, and we have the theorem· If a, 'll8Ct I, prove that the geodesic curvature of the curve cp=oonel ie given by yV'-VrV •,= (Vcp)•--•


l 8, Prove that • fallllly of geod81llce


oharacteriaed by the property

n • ourl t = 0, t being the umt tangent. Deduce that t lS the gradient of some eceJe.r function and tha,t the curves t=oonet. a.re the geodes10 pe.rs.llels to the family of geodesics, meo.aunng the a.ctual geodemo d1Ste.noe from & fixed




19. Show the.t the equation of the md.Jc&tnx e.t e. p01Dt is (r• VD)• r= - 1, the point itself bemg the origin of poeit1on veotol'B.




SIO. Prove that the second curvature JS given by the formula.

2K•(V1 r)'- ll (curlgre.d.:)•. 11,11,1

Sil. Ifp...,r•n,ehowtba.t



v•p~'fl(2K-.l')-J-r•VJ •'fJ(2K-.l')+J-divJr Hence, 1n the oase of a nunllll&l surface, v•p='l.pK. QSI. Prove the relations div curl ct,V=V • v x vct,+ct,v • v x V, vt(ct,V)-ct,v•v+2vct, .vv +vv•ct,. TRANSFORMATION OF ll!TEGRALB

1!3!1. Divergence theorem. We she.II now prove various theorems connectmg line integrals round a closed curve drawn on the surface, with surface integrals over the enclosed region. These a.re ane.logous to the three-parametric theorems of Gauea, Stokes and Green, and others deducible from them. Let O be any closed curve drawn on the surface; and at any point of this curve let m be the unit vector tangentie.! to the surface and norme.! to the ourve, drawn ou//ward from the region enclosed by 0. Let t be the unit tangent to the curve, m that sense for which m, t, D form a right-handed system of umt vectors, so that m=txn, t=nxm, n=mxt. The sense of t 1s the positive senee for a descnpt1on of the curve. If ds is the length of an element of the curve, the corresponding

Fig 27.

displacement dr e.long the curve in the positive sense is given by dr=tds. Consider first a tra.nsformat10n of the surface integre.l of the divergence of a vector over the region encloeed by 0. The area




dS of an element of this region is eque.l to H dudv. If the vector F


given by

F=Pr,+Qr,+Rn, then by (7)




1 [ a,;(:11-P)+a,; (HQ) -JR d.ivF=H and the definite mtegral of d.iv F over the portion of the surface enclosed by a is .,

ff a..v P'dS= ff[~(HP)+!(HQ)] dud•-!! JRdS

N, M being the

= j[HPi;dti+ j[HQ]: du-ff JRdS, pomts m which the curve v = conet. meets 0, and

B, A those m which u = conet. meets it. If now we 8'1sign to du at the points B, A and to dv a.t the pomts N, M the values corn,. spondmg to the passage round O in the positive sense, the above equation becomes

ff divP'dS= f 0 HPdv-f0 HQdu- fJJRdS. Consider now the line mtegral


F • m ds taken round O in the

p0&tive sense. Clearly Rn• m = 0, and mds = t x nds = dr x n.


J. P'•mds= f. (Pr,+ Qr,)• (r du+ r,do) x (",.~- r,). 1

In the mtegrand the coefficient of du is 1

H (Pr1 +Qr,)• (Fr1 -Er,)= - HQ e.nd the coefficient of dv is

1 '1i(Pr1 + Qr,)•(Gr1 -Fr,)=HP, so that

f. F•mds= f. HPdv- f. HQdu.

Comparing this with the value found for the surface integral of the divergence, we have the reqU1red result, which may be wntten




ff JF,ndS ........





This is analogollB to Ge.uss's "divergence theorem," and we shall therefore refer to 1t as the divergence theorem. The last term in (24) has no counterpart in Ge.uss's theorem, but ,t has some important oonaequences in geometry of surfe.cea, and in physical problelll8 connected therewith. From this theorem the invariant property of div F follow, immediately. For, by letting the curve O converge to a point P inaJ.de 1t, we have for the.t pomt



divF+Jn•F=Lt- dS

........... (25).

Now the seoond member of this equation, e.nd also the second term of the first mem her, a.re clearly mdependent of the ch01ce of coordinates. Hence div F must e.lso be mdependent of it, e.nd is thua an invariant. Thie equation may also be regarded as giving an alternative definition of div F. ll23. Other theorems. From the divergence theorem other importe.nt tre.naformations are ea.sily deducible. If, for instance, in (24) we put F = q,c, where tf, is a scalar function and c a. oonsta.nt vector, we find in vutue of (17),


fJ"'cf,•cdS= 4>c.mds-ffJ4>c•ndS. And, since this is true for all values of the constant vector c, 1t follows that

JJ-\ltf,dB=f.tf,mds-JJJq,ndS ........... (26). · This theorem b ... some important applications, both geometrical and phyB1cal. Putting tf, eque.l to & constant we obtain the formula


[, mds= JJndS .... . . . .. .


If now we let the curve O converge to e. point inside it, the le.et equation gives

Jmds .................... (28).

Jn= Lt 'dS

Hence we he.ve an alternative defimtion of the first cu;.,.a.twre of a surface, independent of normal curvature or prmc1pe.l directions. We may state it:

122, 123]



The limiting value of the line integral

J•mds, per unit of enclosBd

an-ea, ,s normal to tJu, aurface, and ite ratio to the unit normal is equal to tJu, forat curvature. In the case of a closed surface another important result follows from (27) For we may then let the curve O converge to a pomt outside it. The line mtegra.l in (27) then tends to zero, and the surface integral over the whole surface must varush Thus, for a cl08ed aurface,

the integral bemg taken over the whole surface. In virtue of (13) we may also wnte this

Jfv2rdS=O. c

Agam, apply the divergence theorem to the vector F x o, where a constant vector. Then by (19) the theorem becomes



JJ curl FdS = c •f. m x Fds- c •JfJn x FdS.

And, smce this have


true for a.II values of the con,tant vector c, we

Jf cw-I FdS = J. m x Fds - ffJn x FdS

...... (29)

ThlB important result may be used to prove the in~ariant propmy of curl F. For, on lettmg the curve O converge to a pomt ms1de it, we have at this pomt curl F +Jn x F

= Lt

J• md.S Fds .........(30). X

Now each term of this equation, except curl F, is independent of the choice of coordmates. Hence curl J' must also be mdependent. It is therefore an mvana.nt. The equation (30) may be regarded as givmg an alternative defirutrnn of curl J'. In the case of a minimal surface, J = 0 Thus (26) becomes

ffvtJ>dS = f. 4>mds, w.





e.nd from (27) we see tha.t

J, mds=O for any closed curve drawn on the surfu.oe. S1mila.rly (29) becomes

JI curl FdS = J, m x Fds. In pe.rticula.r If we put for F the pos1t1on vector r of the current point, since curl r= 0 we obta.in

J,rxmds=O. This equation a.nd the equation


mds = 0 a.re virtually the eq11&-

1Jone of equilibnum of a thin film of consta.nt tens10n, with equal pressures on the two Bides. The one equation expresses that the vector sum of the forces on the portion enclosed by O IS zero, the other that the vector sum of the1r moments about the 01-igm vaoishes•• .Ane.loguea of Green's theorem• are eaaily deducible from the divergence theorem. For 1f we apply this theorem to the function )• mds= Jfw,1.,,,._,i,V•tf,)dS 0

• For the apphoa.hon of. these theorems to the equillbn~m

bra.nest and the ft.ow of heat; m refured to, f§ 15-17.

II ourved

.. (33).

or etretohed mem..

Iamma, see the author'1 paper already




If in tins formula we put V' =conet., we obtu.in the theorem

f.m •Vds=fJv•tf>dS ............. (34),

which could also be deduced from the divergenoe theorem by puttmg I' - V, Geodes1e polar coordinates and the concept of geodesic distance may be used to extend thlS theory m various dll'l!ctlons•. But for tea;r of overloadmg the preeent chapter we ehall refrain from domg this. 1114. Circulation theorem. Con&1der next the definite integral of n • earl I' over the portion of the eurface enclosed by the carve O (Fig 27). If, as before, 'I' =Pr,+ Qr,+ .Rn, we have in virtue of (12) n • carll'= il{~(FP+ GQ)-!(EP+ FQ>}· Henoe, smce i/,S = H dudu, the delhute integral. referred to


Jfn•V x l'i/,S= JJ{~(FP+GQ)-!(EP+ FQ)}dudu

= j[FP+GQi: dv-J[lcP+FQ]; a,,.. If now we ase1gn to dv at N, M and to du at B, .A the values corresponding to the passage round Om the positive sense, this becomes

fJn ,V xl'i/,8-J (FP+GQ)dv+ j (EP+FQ)du. 0


But the line mtegml


J.r,dr= 0(Pr1 +Qr,+Rn)•(r,du+r,dv)

=J.urgh, Vol. (1998). '




magnitudes for the system of surfaces may be defined by the equat10ns a =r11• b == r11, c= r11, /=r1 ■ r1 , g=r1 ■ r1i h=r1•r1, the suffixes 1, 2, 3 having the same moo.rungs a.e m Chapter XI. In terms of these quantities the unit norme.ls to the parametric surfaces are

Expreseions are then determined for the three-paremetric gradient e.nd Laplaman of a. scalar function m space, a.nd for the threepa.rametnc divergrmce and ourl of a vector function. A formula is also found for the fir/II curvature of any surface ,t, (u, v, w) = const. a.nd the properties of the coordinate surfaces a.re examined m some detail. The mtereections of the parametric surfaces constitute three congruences of curves, whrnh are studied a.long the lines explained m the following .Art. Lastly, a. simple proof of Ga.usa's Divergence Theorem ia given in terms of the oblique curvilinear coordmates, l !29, Oongraencea of curve■, The method of Arte 103-5, in which e. congruence of curves 1s defined as the mtereections of two two-pa.remetnc fam,hes of surfaces, 1a not very effective. The author baa shown• that a. curvtlinee.r congruence ia moat adve.nt&geously treated along the same Imes a.e a rectilmee.r congruence. Any surface cnttmg a.11 the curves of the congruence 18 ta.ken ae d,reoto1· /J'IJ/l'face, or surface of reference. Any convenient syatem of curvihnee.r coordinates u, v on thia surface will determine the individual ourves of the congruence, and the diata.nce s along a ourve from the director aurface determines a part1cular point r. Thus r ia a function of the three parameters u, v, s, or r= r(u, v, s) a.nd the fundamental magnitudes a, b, o,f, g, h mtroduced in the preceding Art. are age.in employed. 11 On Oongruenoee of Ourvea." X8hoh MaChamatioal JoumaZ, Vol 28 (1927), • pp. 114-126.

128, 129]



By this method the existence and propertles of the foci and fooal aurfaG6 a.re very easily estabhshed, the equation of the focal surface bemg [r,. r,, r,] = 0. Moreover, corresponcting to the developable surfaces of e. recttlmear congruence, a.re here mtroduced whe.t may be called the envelope 8Urfaoea of the congruence. The number of these to each curve IS equal to the number of foci on the curve. Hitherto nothrng was known of pomts on a curve correspondmg to the limits of a ray in a recttlmee.r congruence. The existence of such pomts on e. curve IS here proved by the following method FllSt 1t 1B shown that

Of all the normals at a given point, to the our11e of the congruenee thr()Tl,/Jh that point, two a,·e also normals to eonaecutive Cllrl/88 It 1B then e.n e&BY step to the theorem :

On eaoh cur11e of the eongruence there are eertain points (called "limits") for whieh the two common nor,nala to tl,,'1,// curve and eonaecutive cur11ea are eoincident. and the feet of theae normals are etationwry at the limit points for variation of the conaeCll!iV8 ci,,rve This theorem then leads directly to the defimt1on of prinevpal aurjaoes and prinevpal p!uJnes for a curve The d,vergenee of the congruence IS then defined as the threepe.rametric divergence of the unit tangent t to the curves of the oongruence. The surface dJ.vt=O may be called the 811.rface of striction or orthoeentric aurjaG6 of the congruence. It 1s shown to have important properties, bemg the locus of the povnts of striation or ortkocentrea, which are the points at which the two common norme.le to the curve and consecutive curves a.re at right angles The orthocentre of a ray of a rectihnee.r congruence 1s the "llllddle pomt" of the ray. The properties of aurjaceB of the congrusnce (Art 104) are examined m some dete.il; and an expression is found for the first curvature of the surface v= ,J, (u), or t (u, v) = const. In terms of the funds.mental magnitudes the necessary e.nd sufficient condJ.t10n that the congruence me.y be normal is


= g,- J.,



which for a rect1hnear congruence is e1mply ./2 = g,. The first curvature of the surfe.cee, wh.tch are cut orthogoneJly by the curves of a normal congruence, 1e g, ven by J=-d1vt, or, if p denotes the value of the product [r,, r,, r,],



The common focal ,ru,r/M• of the congruences of parametric curves, for the tnple system of the precedmg .Art., 1s given by p=O.

EXAMPLES XVII l. If, with the notation of Chap I, the one-para.metno opera.tor V for

a ow'Ve m epa.oe


defined by

V•t=O, V•D=-ic, v .. b=O, vxr-o, vxt=Kb, Vxn=-rn, Vxb=-rb. Also oe.!cul•te the one-para.metric d1vergenoe and curl of ct,t, ct,n &nd ct,b.

provethat V•r=l,

g_ If, for & gmm surface, land m are defined by



show that I, m, D form the rdmproaal system of vectors to ri, r 1, n, satisfying the relations l ■ r1 .,.1,

and Prove &!so th&t

e.nd aimi!&rly that &nd show th&t

m ■ r1=l,

l•r1=m•r1=l•D-=m• D=O H 21-=0r1 -Fr,.., H 1 m=.Er9 -Fr1 ,

r1-EI+Fm, r1 -Fl+Gm,

1'=0/H', m•=E/H', lo m= -Ji'/D', (1, m, D]=l/D. 3. In terms of the vectors 1, m of Ex 2, ehow that Vct,=l~+m~,

eo that 1-vu, m=Ve 4. Prove that the fooe.! curve of a famtly of ourvee on a eurfaos (Art. 126) 18 the mt.t1slopa of the f&mtly, bemg touched by each member at the fo01 of tha.t curve.




5. Show that the focal ourve of tho fe.mtl.iee of pe.ra.metnc ourvea IS also the fooa.l ourve of the f&mtly (u, •)=coost.

6. Prove the.t oonseoutive pa.rametno curves w-oon.et. on a. surface ae.n poesess


oommon normal only where

M~)-o, and deduce t.be eque.tmn div &-=0 for the hne of atnction of the farmly. 7. Show tha.t the f001 on a genera.tor of e. skew surface a.re two unagtn&ry pomte eqwd1eta.nt from the central point, or pomt of etnctJ.on. Prove that, aa the eveolfio ourv&ture of the surface tends to


these pomte tend to

co1nc1dence, and deduce the dU&l. na.ture of the edge of regreamon of a de&a formed by the coa.lesceooe of the focel curve with the

velop&ble aurf&ee, lme of etnctJ.on.

8. The p&r&metno curves a.re orthogone.l, and the curves t.1-=const. a.re geodeBJ.os (chvb=O). If a ourve O outs these at a v&r1able e.ngle 8, 1ta umt t&DgeDt IB e. ooe 8 +b am 8, and 1t.e geodes10 curvature 1B div (b oos 8 - a sin 8). Show that thIB latter 0%presa1on ,. equa.J to -(ain8dive.+~), where d,8/da ,. the aro-rate of more&Be of 8 &long O. Deduoe the theorem of Art. 126 on & f&inily of gecdee1os

9. An orthogonal eyst.em of ourves on a surface is inclined at a vanable

angle 6 to the orthogone.l pe.rametno ou1'Ves. Show that the unit tangents to the curves &re (e.coo8+bmn8), (bcoo8-e.em8) Deduce the value of the tangential component of the veotor Olll'V&ture of this ortbogou&l system, a.nd &how tha.t 1t may be upresaOO. in the alternative forms -(e.diva+bdivb)+D >1rax r1+1Par1 xr:11, s.nd ehow that this vector 1s p times the ra.te of oha.nge of tf, in the d.Jrect1on of the narmaJ, where p=[r1, r1, r3] 1 !;I. If the po1nt1on vector r of & pomt m epaoe 1s a function of the three para.meters u, 111 101 while a., b, 01 I, g, h are the m11.gmtudes of Art 128, and .d, B, C, F, 0, Ha.re the co.factors of these al.em.ants 10 the determinll.llt .D= a. h g =p•, h b f g I • prove tb&t pr11>era=-.Ar1+Br1+Gra, with two &mtla.r formulae. Also show th&t the umt normals to the pa.ra.metrJ.o

surfaces &re r 1 x r 8/.f.i.1 eto. l 8. With the notation of Ex. 12, if 1, m, vectors to r 11 r1, r 8 defined by

m=r8 xr1 1



the r,m.procal system of

D=r1l(r1 1



show that

l•r1=m•r1=n•ra=l 1

while Prove that

l•r1=-m•r1=etc =O.

r -a1+hm+gn, 1

Dl=.Ar1 +Hr,+&r,, a.D.d write down the oorresponcbngformulae for r 51 , ra, m and n. Also show ths.t

l'=.A/D, m'-B/D, n•=O/D, m•D=F/D, D•l=0/D, l•m=H/D 14. If, Wlth the same notation, the three-pa.rametrio V J8 defined by

v-l~+mf,+n~, prove that




V•(Xr 1 +Yr,+Zr,)=p a,;(PXJ+a.;(pY)+ilw(pZ) , Vx

(Pl+Qm+Bn)=!:.:i: (R,-Q8) l'J, I'

Deduce from the former that the first curvature of the pan.metrio surface 'lli-oonBt, 1s given by

-~[!(A-)+:-(~)+!(!?.~)]· °" ow ./.A

J1' ,J .A "" ,JA, Also prove the identity V • vxF-0, where F


&ny veator pomt-funotJ.on.

129] 15.



a, b, a,f, g, ha.re the magnitudes of A.rt 128, prove the relations

r,,r.,=ib,, r1•r,.-h,-!bi, r,,r,.=i(u,+h,-/,), nnd Wl'lte down eJ1 the correspondmg formulo.e I 8. With the not.&llon of Eu 12-16, show th•t, for a triply o,lhogonol of surfaoes (f=g=h=O),




F-G=H=O, B=ca,


l=r,/a, m=r,/b, n=r,/a, V=!!!_+ ~! + !! !_ a OU





The first order magnitudos for the surface tt=oonet e.re b, O, c; and the second ordsr wogtutudes ore - bJ2./a, O, - c,t2Ja. The first curvature of the Surf&ee JB

!J.-.• ../abcau


The second d8l'lvafaves of rare given by

r11-½(a,_l-a,m-a,n), r.,-!(b,m+a,n), and s1mllar formu.Ia.e, and the denve.t1ves of 1, m, n by

11 =

-i., (a,_l+a,m+a,n),


-i., (a,l-b,m),

and so on. La.m,'s rela.tions are equ1 valent to

~-1:-~a+a~•• ✓ai[f.(~)+~(~)J+~-o, with eim1l/U' formula.e

1 7. W1th the not.&tion of .Al't. 129, show that aonseontive ourves of the oongruenca Mll meet only where [r1, r 1 , ra]-0. This is the equation of the l'ocal aurfac~.

l B. For any ,,..eo,, poim-ju,nct.ian m ,paos, Iha ,colar wipZ. produ.l of it, derwatil/88 "' th,,. non-capla,uzr direot,an,, divU.tl by th, ,calar tnpZ. prodwe of tha unit wotora in those d,recti"ona, i, an in:variaw.t. 1 9. For • family of parallel surfaces, the tbre&-parametrio funollon curl D vamsh&B 1denticaJly. !10. If a family of ocrvss on • surface outs • family of geodesios at &D 1.ngle which is oonatant a.long &ny one curve, the geodesic ourva.ture of any ,iomber of the former V&Ill8he., at the line of stnot1on of the !attar. (Ex 8) ~




130. l'e.mll:, or curve■ (continued) Some further important p1operties of fam1hes of curves on e. surfe.ce should here be mentioned. Consider first the arc-re.te of rotation of the tangent plane to the surface, as the point of contact moves e.long one of the curves We have seen that the d1rect1on of the a.xlB ofrots.tion is the d1rect1on conJuge.te to that of the curve at the pomt of contact (Art. 35). The author has shown• that If t ,. the unit tangent for a, fum,ly of CU1'11S8 on a B1J,rface, the tangential oomponsnt of curl t gives both the d,recti,on of tns amis of

rotation of th• tangent plane, and tJ,e ma.gnituds of the a.ro--ra.te of turning, ae the point of contact moves along a. etW'l/6 oj the family. In the case of e. family of geodesics D, curl t van1Shea 1denticsJly, and curl t 1s therefure tangent1s.l to the surface. Thus:

JftlB tnBf//111ttangentfora.fonn,!yof geodesiC8, cu,·l tgi1J111boththe direction conjugate to that oft, and also thB arc-rats of rotation of the tangent plans, a.a ths point of contact mo"88 a.long ons of th, geodSBios. The momsnt of " family of curves may be defined as follows. Consider the tangents to two consecutive curves a.t two points dIStant ds along an orthogonal tr&Jectory of the curves. The quotient of tbBlr mutual moment by ds' 18 tlie moment of the fa1111ly e.t the point COllSldered. It is a point-function for the surface; and the author has shown the.tt thB moment of a fcimily of OU1'11S8 with unit tangent t has the valu• t, our! t. ThlB 1s equal to the to1s1on of the geodesic tangent, nnd vanishes ,vhsrever a curve of the family 18 tangent to a hne of curvature (Art 49). The locus of euch points may be c&lled the line of eero momsnt of the family. Its equation 11 t • curl t = O. Similarly the !ins of normal Cll1'lla.tlwe of the family is the locus of pomta at which their geodesic curvaturo is zero Its equation 18 n • curl t = 0 And the line of tangsntia.l cu1'111Jture is the locus of pomts at which the normal curvature vanishes. It is given by n x t , curl t =0. In connection With a fe.lillly of parallels the author h1LS proved the theorem:, A fl8088Bf1,ry and B1J,j/ioient condition tna.t a family of CU1'11S8 with unit tangent t bB a.family ofpa.ra.llels is that d,v t vanlBh identically. •


0n Families at Ourvea a.nd Smfaoel." Qua,wrlg J'ounaal o/ Maihemanc,J

Vul &O (19i7), pp 810----381, t Loe. ed., § 6.


c11. 1 § 7.




The quantity div t may be called the diwrgtmJJtJ of the family. Thus the cha.ra.oteristic property of & fanwy of p&r&!lels is that it.a divergence is everywhere zero. Further, -chv t IS the geodeaio ourvature of the orthogonal tmJeotol"l.8Z of the family, and, if thlB 18 zero, the orthogonal traJeotonee &re geodesics. Thus: Tll8 orthogonal flra,featonea ofa famwu ofp(J/f"(J,llsl ourt1ea constituta a family of geodesios. And con1J8'1"aely, ths ortliogonal trajeotoM of a family of geod6&1CB conmtmi,e a family of para!!els Thus to every family of parallele there is & family of geodeaice, &11d vies IJ8'l"sa.. The expression "geodeB1c parallels" is therefore t&utolog,.cal, &a a.II parallele &re of tlus nature. .And, in connection with the properties of geodeBice, the following theorem may alzo be menlaoned•: If a fannily of Ollrt/68 on a 11/J/t"faos outs a jami!y of geodmos at a,i angle which 18 constant for Bach ourve, ths !in, of normal our.atun, of the former i8 the line of llflrlf:tion of the latter. With the not&laon of Art 122 we may define thej!ua, of& family of curves e.crose any cloeed curve O drawn on the surmce, &a the va.lue of the line mtegral

f. t •

larly the value of the integreJ

m ds t&ken round that curve. Simi-


t, rl.r may be ea.lied the owouZalilm

of the family round 0. Then from the Divergence Theorem it follows immediately that:

Tll8 tJUr/a,os int,gral of the divergenos of a family of OtWM ~ a,iy region i8 egual to the :ftw,: of the fam'Wfl aoroas ths ba,,mi/,M"fl of

tu region.

Similarly from the Ciron!&tion Theorem we dednce that: Tll8 tJUr/aos integral of tu geodssi.o CUf"ll