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DATE LABEL
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Class No
I
ADVANCED ALGEBKA
ADVANCED ALGEBRA
BY
S. BARNARD, M.A, FORMERLY ASSISTANT MASTER AT RUGBY SCHOOL, LATE FELLOW AND LECTURER AT EMMANUEL COLLEGE, CAMBRIDGE
AND
J. M, CHILD, B.A., B.Sc, FORMERLY LECTURER IN MATHEMATICS IN THE UNIVERSITY OP MANCHESTER LATE MEAD OP MATHEMATICAL DEPARTMENT, TECHNICAL COLLEGE, DERBY FORMERLY SCHOLAR AT JESUS COLLEGE, CAMBRIDGE
MACMILLAN AND CO., LIMITED ST. MARTIN’S STREET, LONDON 1939
QLLPHfi IQBPL LIBRQRY
3441
COPVkIGHT
rKINTHi)
IN
,
In this case, the values of the six crossratios
—CO,
—CO,
—
— co^,
—0)2,
and the system z^, 23, 23, 24 is said to be equianharmonic.
12. Homographic Substitution.
(1) If Zj^, z^, 23,24 are any four values of a variable z, the value of {z^z^, 2^24) is unaltered by the homographic substitution
Mn^rj z^(lZ + m)l{lZ + m).
For if Zj, Zg, Z3, Z4 are the corresponding values of Z, as in Art. 1, (Im' Vm)(Z^Z^) “ (i'Zi + m') {I'Z^ + m') ’ with similar equations.
It follows that {2i22, 2324) = (ZjZg, Z3Z4).
(2) If a crossratio of A, B, C, D is equal to a crossratio of A', B', C', D', then any two corresponding crossratios are equal, and we say that the two sets of points are equicross* [ABCD)=^{A^B'C'U).
This is indicated by writing
•Also called equiaoliaxmonic bj some authors^
IQ
K.AJNfJES AND PENCILS
13, The
If p is
Biquadratic.
mined hi/a, /3, y, S, the roots of then
»
one of the six crossratios deter¬
+l^x^ + Gcx^ + 4f7x + e0,
+ l)"(p  2)(p 
= 27J“(p^  p + 1)^,
.(A)
irhich is equivalent to IJ {p^ p + \f = 27/V ip^f.(B)
For suppose that
,n
c,
{Pa)(y&)
p = (^y,aS) = ^^g~— wliere A = /Sy + a8, p. = yoc + pS, ;^ = a^ + yS. It follows from //,/!., XII,
whore
^2’
10, that p = (^i —
roots of
^3
.l/3_/, +.(C) Let
2
= pHp”\ then using the equations
= 0,
/jVjj =  j;i,
4^1^=III  Jy
it is easy to show that 36^3
9(/^,J)
3J z — \ t\=~^ ' T ^ .
and therefore
Substituting this value for t in equation (C), we find that
P
27J2
p  27J‘^
4(2iy*“(2c5)(c + 2)“27(c2)* Since
2
= p + p~^ and A = P ~ 27J, equations (A) and (B) follow at once.
Hence the following conclusions : (i) If /—0 and are e“, and the points a, /3, y, S
(ii) If J = 0 and 1^0^ then p= 1, 2 or 8 ore harmonic.
and the points a,
(iii) The crossra/ios are all real or all imaginary according as in the first case the points a, For
y, and
y, 8 lie on a circle or they are coUinear.
^2, ^3 are all real or only one is real, accordins; as zl 7!j0,
14. Ranges and Pencils.
A set of collinear points is called a range . the line on whicli they lie is the axis of the range. A set of concurrent lines is called a pencil: the point at which they meet is the vertex of the ])encil.
CROSSRATIO OF A PENCIL
15. Crossratio of a Pencil. y=fMjX,
11
{1) If the pencil of four lines y^fi^Xy
y^p^x
is cut hy any transversal ax\hy\c = 0 at P^y P^y P^y P^y then
For if (xi, yi)y etc., are the coordinates of P^, etc., /p D
and
3/1
DP \
_^1^3 *^2^4 (^1 ~ ^3) (^2 “ ^4) /__\  P,P, . P,P3=’
.. C
%Vo
a f 6/Lti
^1= a + biMs) (a + bui) (a + 6/X3) ’
with similar equations. Therefore {XiX^y x^xf) = » and the result follows. The ratio (PiP^y ^3^4)* which is the same for all transversals, is called a crossratio of ike pencily and is denoted by 0(P^P2y P3P4), where 0 is the vertex of the pencil. If OCPiPg.PgP^) = 1, we say that the pencil is harmonicy and that OPj, OPg and OP3, OP4 are pairs of conjugate rays. (2) It follows that the crossratio of the pencil formed by the four lines OAy OBy OCy OD is given by n/jp ^in\ sin ^00. sin POP* 0{AByCD)= . —. , sin AOD . sin BOC where AOO is the least angle through which OA must be turned in order that it may coin¬ cide with OG {H.A.y V, 11) : and so for the other angles. For denoting the angles xOAy xOBy xOCy xOD by a, JS, y, 8, we have ^ ^ sin(ay) sin ^00 Ml — Mi—tan a — tan y —, ' cos a cos y cos a cos y with similar equations. Hence 0{AByCD)y which is equal to (miM2»M3M4)> has the value stated above. (3) Or geometrically thus : Let p be the length of the perpendicular from 0 to ADy then AG . p = 2A A0G = 0A . OG sin AOG. Using this and similar equations, we have /AT? riTw AG.BD sin .400 . sin POP (ABy LV) = = sin ^OP . sin BOG ’ This is the same for all transversals, and is called the crossratio of the pencily and is denoted hy 0(.4P, OP). • This is frequently taken as the definition of the crossratio of a pencil.
VANISHING POINTS
12
16. Homographic Ranges. (1) Lot the variables x, z'(which may be complex) be connected by tlje hoinographic relation pxx \qx\ rx { — 0.(A) Let the values of z, x' be represented by j)oints JV, X' on the same or on difTerent axes. If (a, a), (6,are pairs of corresponding values of z, x\ the corresponding points being (A, A'), (B, B ), ... , then the ranges (AB... A'...), arc said to be homograpliic.
(A'B' ... A'...)
This is expressed by writing
(AB ... X ...)== (A'B' ... A'') or simply (X) = (X'). (2) It follows from Art. 12, (1), that am/ four 'points of o?te range are equicross with the corresponding points of the other. (3) Conversely, if a correspondence exists such that any four pomts A,n,C, X of one range and the corresponding points A\B\ C', X' of another range are equicrosSy then the ranges are ho}noqraphic* For wo may regard A, A' as variable and the other points as fixed, then, since (XA,BC) = (X'A\B'C'), we have x^h a~c
x'b'
ac
x~c' a~~b~V^'' a~b''
.
and this equation reduces to one of the same form as (A).
17. Vanishing Points.
If p^O, equation (A) can be written r /'
that is to say,
V
p
(D)
V
IX . J'X' = {qr  ps) 'p\
(E)
where I, J' are the points x=  rip, .r'= q’p.
The points /, r are called the vanishing points, and if oo , oo ' aro the y
an_^( s, / corresponds to » ' and J' to oo .
are nn t]
IZli ri
Same Axis.
“■
lose ore tlio
(1) If two hoinographic ranges
»'
corUpcid. 10
> in Roometry, this stntemoni Is taken as the UeUnition of homoRraphIc rang^.
INVOLUTION The origin being the same for both ranges,
13 G are given by
J?x2 + (g' + r)a: + s = 0, .(F) therefore
OF\OG= (q + r)/p = 01 + OJ\
so that FG and IJ’ have the same midpointy
Fig. 6.
(2) If Xy X' are corresponding points, since F corresponds to F and G to Gy by equation (E), IX. J’T = IF. J^F = IG. J'G.
.(G)
We also have such equations as
(FAy BC) = (FA'y BV')y
(XAy FG) = (X'A'y FG).
.(H)
Again, since (XAy FG) = (X'A'y FG), it follows that
(■^X'y FG) = (AA'y FG) = a constant, .
..(I)
(XX', FG) = (loo ’,FG)=^4\
and conversely, if this relation holds, tlic pairs are in involution. [For {. IX. 7LV') = (. I'A". />*'A').] 14. If {A, A'), (7?. 7r), {(\ O') are pairs of points in involution and L, J7. A’ are the midpoints of .1.1', HB', Cr\ prove that
(i) A A '2. MN + 7i/F= . A7. + CV'^, 7.A7 =  4.VA'. XL . 7.A7. (ii)
(ii)
[I^t 0 he the centre of the involution. Denoting O.I. 07.,... we have art 6//or'= 1*. Now prove that A A''^A[l~ h) and
by a / ■ ' »
•
ta n ""•'■I’'’""I"'"' "> involution .lotorminod bv the paim (rt, rt ), (rt. b ), the equation connecting .r, .r' is ' ^ xx'{r, +
 {h I h'))  (.,• + a)
(We have (nx. h.r’)
h'.r).l
 W/) I
I,')  bh'(a+a')=0.
CENTRE AND FOCI OF INVOLUTION
19
16. If (a, a')y {x, x') are pointpairs in an involution of which /, g are the foci, then ax a—x' ®“/ ^ — g a'x a'x' , (.\ ]) are cquianharmonic.
it is required to find a point D such that
(ii) Show tliat there arc two positions (/>,, !).>) of 1) and no more ; also that for each of these DA : DU : DC = \la : 1/6 ; 1/c. (iii) If IS.ABC is named so that UCy CA, AU subtend at Di are . I + 00=’,
B + GO^
Z.J5oints. The rest follows from Art. 23.] 22. Txt u = 0y ?/' = 0 bo equations of the second degree in a:, y ^\ith real coefficients. Let the conies represented by « = 0, ?/' = 0 meet in P, P\ 0, O'. Trove that (i) If one of these points (0) is imaginary, then another ((?') is the conjugate imaginary point. (Art. 6 and //..I., V. 14.) (ii) I^t .4, 7?, C be the points of intersection of {PP\QQ% {PQ\P'Q)y )• Ihcn I, By C are all real xinless two points of intersection (P, P') are real and two (0, O') •'tre imaginary. In the latter case A is real, B and (’ are conjugate imaginarv* points, so that the line PC is real. (Art. 6, Ex. 1.) ^ r' I he triangle ABC is the diagonal jioint triangle of the quadrangle 7^P'0'0» iS selfconjugate with regard to every conic through Py P'y O', 023. In the last e.xample, if the eiuations arc
w2a7/^= + ,r2 = 0.
(’ = a=j/ = 0,
proyc that the coordinates of .1, P, (' aix)
1
T)
4*
4
2±k3
7±k'3I \
and that the equation of BC is 2.r  4f/ [ 5 = 0, Verify that BC is tlie polar of .1 with regard to each of the conics. 24. Find the vcitices of the common selfconjugate whose equations arc a =  2ip i10/y  13  0, showing that the
triangle of the conics
~ S.c + 6// 7=0,
coordinates of these points aix' (1,1), (2,2), ( — 2,V^).
CHAPTER II THE QUADRATIC AND SYSTEMS OF QUADRATICS
1.
Resultant of Two Quadratics.
Let a, P and a', yS' be the
roots of the equations
u ~ ax^ + 2bx + c = 0, respectively.
u' = a'x“ + 2b'x + c' —0
These equations have a common root if and only if
Let
.(A)
R = a^a'^{oc—0, the signs of ^ as x passes from  oo to oo are shown below: X
dx
—
Xi
CO
+
0
+
X.

0
00
+
Therefore x^ gives a maximum and Xg a minimum value of y. (ii) If (a6')
/'=
A3=^Zj ^yZg + cZg;
similar functions of the m*s and n'a ^+ ^3A3, + W2V2 + ^ngVg = Mi/Xj + Wajag + ngjUg,
with similar values for the other coefficients. Hence, by the rule for multiplication of determinants,
h
Ag
h
Ml
M2
Ms
^2
^2
X
h m^
k
k
mg
mg
Wa
n^
s
a h 9 h b f 9 f c
1
X
h
k
k
mj
W2
mg
^1
^2
«3
The function (^1^12^3) is called the determinant of transformation, and on account of the property just proved J is said to be an invariant of w. For the nonhomogeneous function
u = ax^ + 2hxy + by^ + 2gx + 2fy + c, the corresponding homographic substitution is
LZ + miFHni Z3X + TWg y + W3 *
+ ^
+
l^X + m^ Y + W3 *
Suppose that these values are substituted for x, y, and that we multiply by (l^X + mg Y + 7^3)2 to remove fractions. Denoting the result by u\ it will be seen that the coefficients a\ 6',... are the same as before, so that the result just proved continues to hold.
TWO QUADRATICS
36
EXERCISE III In this exercise (a, ^)» (a , u = ax‘^ + 2bx + c = 0, Also
are the roots of
u'=a'x'^^2h'x + c' = 0,
A =^acb^.
Other letters, as J, x^,
A'= a'c' b'^
where a, a' are positive.
K = ac' + a'c~ 2bb'.
Ai. Aj, have the same meaning as in the text.
1. Determine the constants so that 4:x ~S = A {X a)^B{x  b)\ 2. If a 3. K w where
x^ ■~2x  5 = A'{xa)^ + B'{xb)K
prove that x — a is a factor of J. ax^\2bx + c, u'=a'(xa')^, then u=A{x~XiyhB{x  «')S acb^ hoc'he . {aa'hb)^ B= A= X, = act^ + 2hoL + c act^ + 2bct' + c aa' + b*
4. Referring to Art. 7, show that (i)  a'c') (a  A^a') (a  A^a') == {ab'  a'by. (ii) If 6'24aca'c'> 46*6'*.] 5. Draw the curve represented by
(x2)(x3) (X l)(x4)*
[The interval (2, 3) is within (1, 4); but a/a' = 6/6', so that one turning value is at 00; also _ 2 dy _ 2(2x5) ^ ^ X*  5x 4* 4 ’ dx X®  6x + 4 * Hence, x = 26 gives a turning value.
There is no real point of inflexion.]
(xl)(x3) 6. Draw the curve represented by y = (x2)(x4) [The intervals (1,3), (2,4) overlap, /. y has no turning values. negative for (a6') < 0.
Also^ isalways ax There is one point of inflexion lying on the line 2x + 4y=7.]
7. If p = (aj3, a'j8') where (a, )S), (a', )3') are the roots of tt = 0, u'=0, prove that (ac'4a'c266')*
m)‘‘ 4(ac6*)(a'c'6'*)‘
8. Prove that
»/®= (J'u*^ww'4Jw'*) = (6'*  a'c') (m  AjU') (w  Aj«'). Hence show that if 6'*