270 
MR. A. CAYLEY’S ANALYTICAL RESEARCHES CONNECTED WITH 
the equation of the surface, 91, 33, C, JT, <B, H, K as usual, and 
«=j(v/aic+jrv/s+®N/s+®yc), 
then the equations of the required sections are 
(ax-^hy-\-gz)X.-{-y\/ C+2\/33+\/ —ap\/ 1 + X^ifj=0 
^~\~{hx-\-by-\-J^zyY-\-Z\/^-\-\/ — bp \/ 1 -|- = 0 
^\/^-\-y\/9i+igx+fy+cz)Z+\/ —cp\/l+Z^w =0, 
where X, Y, Z are to be determined by the following equations, 
(/+2^yi)+V^(Y+Z)+/YZ-x/^x/T+YVH^^=0 
(g-l-2^\/3!3)“}"\X33(Z-l-X)-|-g'ZX — \X c(i\/ l-\-Z^\/ l-l-X^=0 
(h-\-2d\^ (2!D(X-1- Y)-1-AXY — \/ cih \/ 1 1 -|- Y^=0 ; 
and the solution of which, putting 
f=^3x/ysc-jf, h=4/€WM-% 3=\/i</m€, 
is given by the equations 
KX=®+(-f+g+hf-2(-f+g+h)J 
KY=?®+(f-g+h)*-2(f-g+h)J 
(f+g-h)"-2(f+g-h)J* 
Instead of the direct but very tedious process by which these values of X, Y, Z 
have been obtained, we may substitute the following a posteriori verification. 
We have 
K'(l +X=) =4{-f+g+h)y’(l +^) (i -f) (i -^) 
KVn-YVl+Z’=4(f’-(g-h)")j’(l-j)\/ l-fl\/ l-p 
K"{1+YZ)=4(^1— gh)(p— (g_h)")— 2gh(g— h)*} 
K(Y+Z)-2f"-g''-2h*+4j’=4(l-j)(J'-gh). 
Putting also 
f'-g'-h=+^-^’=(p-(g-hf)- 
2gh(J2-gh) 
J2 
K^= 
(f»-{g-h)=)({g+hr-f’ 
4g^h^'^ 4g%^(g— h)® 
) P ’ 
It is perhaps worth noticing that the value of the quantity A previously made use of, is 
