258 
MR. A. CAYLEY’S ANALYTICAL RESEARCHES CONNECTED WITH 
And in order to verify the geometrical construction, it only remains to show that each 
bisector touches two tactors. Consider the bisector and tactor 
2/3 
(i +i)3/+ (^^-2^0^+ (^“/3)^-o^ 
— 2a(aa— >/ — 2bc)x-^{c-{-a)y-\-{b-jra)z-\-{2aa — \/ — 2bc)w=(^\ 
and represent these for a moment by 
}!x-\-^y + v'z-^^w = 0 . 
If A be the same as before, and A' the like function of X', (/J, v’ , also if 
d> = xx' + (Mz/oo' -|- vv' — + (ju'v) — (i-x' + ^'x) — (Xfztz' + X'(M/) — 2§§', 
then 
^'^—{2ao^—2cts/ — 2bc-\-b-\-cy, 
O —act ( 2 +^) —2ci^ —2bc(2-\-—^-{-c(2-\-j^ ; 
and the condition of contact AA=0 (taking the proper sign for the radicals) be- 
comes 
(2+i)(2««'‘-2^.^/^:2K^-i+c)=aa“(2+i)’-2«^/^^(2+;;^)+c■(2+i); 
or reducing, 
«a— &/3-|-c^^^_^l = 0, 
an equation which is evidently not altered by the interchange of a, a and b, jS. The 
conditions, in order that each bisector may touch two tactors, reduce themselves to 
the three equations, 
aa b^-\-c 2 ^^ ^ j b, 
^ 2|3y -1- 1 ^7 — Oj 
which are satisfied by the values above found for the quantities a, Z>, c. The possi- 
bility and truth of the geometrical construction are thus demonstrated. 
§4. 
Let it be in the first instance proposed to find the equation of a section touching 
all or any of the sections x=0, y—0, z=0 of the surface of the second order, 
ax^ + by"^ -b cz^ + 2fyz -j- 2gzx -f 2hxy -\-pw^ — 0 . 
Any section whatever of this surface may be written in the form 
{a\+h[x -\-gv)x-\- (AX-b bg-{-fv)y-\- ( gX +fg-{-cv)z-\- s/ — p'^w = 0, 
