268 
MR. A. CAYLEY’S ANALYTICAL RESEARCHES CONNECTED WITH 
and reducing, 
•yy 
K2@ 
Also 0=2X0 gives 
o= 
Suppose 
V \/2B€ ) 
K@ 
then substituting 
that is, 
2x/( ^/n€ + (^){ + \/U€ 
\/BC+4r=a, \/^C~4r=a; aa;=Ka 
^m+0=(i, ii(5,=Kb 
\/Qi5+|^=y, i?=y, 77;=Kc; 
0_ V z>cO=o 
y^i7, 
0-^(« + aj('l 
/3/7/ 'v/a + a/ ’ 
5+?+0{>I+2«l) — -/(^+20|) } =0 
which may be written 
where 
L|+M^+N^=0 
L'H-M';?+N'^=0, 
S^2 Vc 
L=i+2^‘- - - ;^+“' (yg-2y), M=^+ N=i -'^y“+* ' 
L'=l+2^" 
^ /3;7; 
4ic(« + a^) 
'^/3y7/ 
^/7/ 
N'=l. 
Yi 7-^), M'=^ 
HiYi \ v^a + a^/ 
Or since |, ??, ^ are equal to 1, Y+Z, YZ respectively, 
1 : Y+Z : YZ=MN -M'N : NL'-N'L : LM'-L'M 
4Sc(a + «,) 7 
' /3,7, V 
1- 
2 y' a-\- 0.1 
^ 1^/7 1 
4bc{a + a^/ V'2\7o + «i 
ftr, V+ ■'Jv v^)+ (Z+Va). 
\7 2 s/ a + 
Also 
/+.ya=|=^, 
