228 
ME. A. CAYLEY ON THE POEISM OF 
as determined by the single parameter^. And a point of the conic UH-W=0 may he 
considered as determined by the same parameter ^ which determines the tangent at that 
point. 
As regards the conic V=0, the common parameter for all its tangents is oo, and we 
may consider any other tangent of this conic as determined by the parameter and a 
point of the conic as determined by the same parameter 0. 
Suppose, in the first instance, that the two conics are 
Y= z^=0; 
the equation of the tangent of U+W=0, the parameter whereof is^ (in fact a com- 
mon tangent of the conics U-|-W=0, UH-pV=0), is easily found to be 
^\/h — C\/ (L-\-Tis/ c — ct — h\/ Oj 
and if this meet the conic V= 0 in the points P, the parameters whereof are co, 
and cx), & , or say 0 and 6' respectively, then the coordinates of the point P are given by 
x\ij\z—s/h—c\/ c—a\/b-\-6\s/ a—h\/ c-\-d; 
and substituting these values in the foregoing equation, we have 
{h—c)s/ a-\-ks/ a-\-^-\-{c—a)s/h-\-k^b^p^b-{-6-\-{a—h)^c-\-k\/c-^p^/c-\-& 
as an equation connecting the parameters jp and 6. This equation may be replaced by 
(iX-j- 9) — 
\/ [b-{-k){b +p)(9 
^/ {c-k-k){c-k-p){c-\-b)—'k-\-yjC, 
from which X, jO/ are to be eliminated ; and squaring and reducing, we have 
Y=zabG-\-kp^, 
—2Xyj—bc-\-ca-\-ab—{pb-\-kp—M), 
= 0 / -|- b -}- c “1“ k ~\~p T" 9) 
and thence 
(bc-\-ca-^ab—p&—kp—k()y'—4L{a-\-b-{-c-\-k-{-p-\-6)[abc-\-kp6)-=iO 
as the rational form of the original equation. But the same rational equation would, it 
is clear, be obtained from the system 
\/ {k-{-a){k-\-b){k-{-c)=L-k-Mk, 
s/ (2?+«XP+^)(P+^)=L+Mp, 
s / “b “h “b c)— 
by the elimination of L and M. And it follows from Abel’s theorem (but the result 
might be verified by means of Eulee’s fundamental theorem for the addition of elliptic 
functions), that if 
