256 
MR. A. CAYLEY’S ANALYTICAL RESEARCHES CONNECTED WITH 
{h-\-a)iJij — {c-\-a)v; or since the ratios only of the quantities k, (l, v, p are material, 
[j!j=c-{-a, vz=b-{-a, and therefore 
A^=X^—2(2a-\-b-\-c)K-\-{b—cy—2f=(K—b — cy, 
or y= — 2{aX-{-bc). 
Whence the equation to a section touching bx—ay=0, az — cx=0 is 
'KX-\-{c-\-a)y-\-{b-ya)z-\-'s/ — 2{aX-\-bcy\w=Q. 
And to express that this touches the determinator in question, we have 
— b — c) ^ — (ji{2ci-\-b-\- -j- 2\^ — 2 {cTk -|- 6c) \ 
and selecting the upper sign, 
2aa=— 2\/ — 2{a\-\-bc) ; 
whence 
?v= — 2oi{aci—\/ —-26c), \/ — 2(a?i-j-6c) = (2aa— \/ — 26c) ; 
or the section touching the determinator and the sections bx—ayp=0, az — cx=0 is 
~2ot,{aa—\/ —2bc)x-\-{c-\-a)y-\-{b-\-a)z-{-{2ao(,—\/ — 26c)zc = 0; 
and at the point of contact with Ihe determinator 
1 1 
2yz-\-2zx-\-2iy-\-t(f=Q. 
Eliminating w between the first and second equations and between the second and 
third equations, 
— 2bc(ax-\-~y-\-Yj)-^cy-\-bz=0, 
and from these equations {cy—bzy=0, or the point of contact lies in the section 
cy~bz = 0. It follows that the equations of the tactors are 
— 2a{aa — ^ — 26c)x-{-(c-\-d)y-y{b-\-a)z-y(2aci—Ay —2bc)w~0 
(c-\-b)x—2^{by — yy — 2ca)y-\-{a-\-b)z-y{2b^ — ^ — 2ca)'ic = 0 
{b-\-c)x-\-{a-\-c)y—2y{c'y — \/ — 2ab)z-\-{2c'y—->y ~2ab)iv=0, 
where a, 6, c still remain to be determined. 
Now the separators pass through the point of intersection of the determinators ; 
the equations of these give for the point in question, 
X :y : z : w=:{2^ry-\-l)( — a-\-^-{-y-\-2a(i'y) 
: (2'ya-y l)(a — l3-\-'y-\-2a(3y) 
: (2a|8-|- l)(a-j-(3 — 'y-{-2a(B'y) 
4a^j3y-]-ya^-\-(B^+f; 
