264 
MR. A. CAYLEY’S ANALYTICAL RESEARCHES CONNECTED WITH 
But since 
are proportional to 
it is only necessary that 
should be proportional to 
{cc' + rW, («"+7")^3" 
7 
m 
J ’2 
Or since the equations are supposed symmetrically related to the system, we must 
have the second set of relations between the coefficients. Suppose 
^2 + ,y2 _ 82 ^ ^'2 ^ _ g/2 ^ _j. ^112 _ §;/2^ ^ 
y2 — ^2 y/2 — ^/2 y/2 — ^IIZ ip’ 
then since 
— a^= — 4(y+^j3)(pj8, &c., 
we have 
=/3" +y" — 4.s(y +9/3)^ 
=/3'^ +y'^ -45(/ + 9|3')i8' 
^//2 ^ 1^,12 ^ ^112 _ 
and I” will be supposed henceforth to satisfy these equations. 
We have next 
A'2 -a^=4(y'+cp(i')(3'{s+<p+Z-{-sZ^) 
A"2+B"^-C"^=4(/+?)/3")|3''(5+9+Y+5Y^), 
which may be simplified by writing- 
ft — <5 l+/t(p 
1+1^’ F- — f 
where [ji>, v are to be considered as given functions of s and <p. These values give 
A'^ +B'^-C'^ =4(y'+(p(3')(5's(Z+fx.){Z+u) 
A"^+B''^-C'^=4(y''+^(3'')(3"s iY+u.){Y+v). 
Hence, putting for simplicity 
P=4(y'+p|3')(r"+®(3»)(3'(3", 
we have 
4{Z-\-[/j){Z-\-<'){Y +t') = U^+/i:[(aH-/3(Y4-Z)+yYZ)^ — ^^(1 d-Y‘)(l +Z’)]. 
And the two sides have next to be expressed in terms of Y+Z and YZ. 
If for symmetry we write 
1=1, ,=Y+Z, ^=YZ, 
then 
— 0^+^^] = U^H-A-(aH-/3;7+yQ*. 
And U is now to be considered a linear function of |, 
The condition that the first side of the equation may divide into factors, gives an 
equation for determining k ; since the condition is satisfied for ^=0 and k= oo, the 
