984 
PROFESSOR A. CAYLEY ON THE SINGLE 
Also considering two other functions & 0 (w) and 9- n (u), or as for shortness I write 
them, S () and d 13 , we have 
V=aX+/3Y+yZ +S W, 
V=*X-£Y-yZ+SW, 
and substituting the foregoing values of X, Y, Z, W, we find 
Md 0 3 =A^+BY 2 +Cz 2 ,+D< 
Md 13 2 = Car + D y ~+As 3 +B w 2 , 
where writing down the values first in terms of a, ft, y, 8 and then in terms of the cs, 
we have 
M= 
(a 2 S 3 ) ((3~ y 3 ) 
> 
1 
OO 
O 
ii 
A= 
2 . 2 
3 3 ^2 ^6 ’ 
B= 
— aS (/3 : — y 2 ) + /3y (a 2 — 8 
0\ 9 0 
“) = » W“ ( 
C= 
a 2 /3 3 —y 2 S 3 
o 0 
= 3 3 CPV, 
D= 
— aS(/3 2 — y 2 ) —/3y(a 2 — 8 
0\ 9 0, 
') 3 3 C 15~ C h 
and we then have further 
that is 
q-V —c 0 c 13 -9 4 d- 8 -f" t *3 c io'^7'^in 
C 4 C 8^0^12 ==C 0 < h2 a - 2 ' _ i" < -'3 C 15 J W > 
whence equating the two values of d 0 2 V 2 we have the required quartic equation in 
a:, y, z, w. 
123. But the reduction is effected more simply if instead of the c’s we introduce the 
rectangular coefficients a, b, c, &c. We then have 
M = ( c "~-b' 2 ), A=-a"c, C =ab, 
B=~b'c-b"c", = be ; D=&'&"+c'c" l = a'a", 
and the equations become 
{c"~ — b' 2 )9- 0 2 = —a'ex 2 -\-bcy~-\- abz^—da'w*, 
(c" 2 - Y 2 )V- abx 2 - a'a'Y-ct'cz 2 + bevr 
\/b c d (J d 1 o =: \/ft xz-‘—b c yw> 
so that the elimination gives 
o 
