MR. A. CAYLEY ON THE ANALYTICAL THEORY OF THE CONIC. 
661 
or we may assume 
±x/(A, . .XX, (M, v) V(A, . -B', . -B, 0, ^'=K, 
so that the conic 
{±\/(A, . -lA, >)V (A, . -Ia', f/.', f/-', OlK •••& 
breaks up into a pair of lines. 
Putting for shortness 
+ x/(A, . .Ja, vf\/ (A, . .J}!, yJ, v'y — {A, ..X^, )^X^', v') = Q,, 
the coefficients on the left-hand side of the equation are 
(Q*+2KU', Q/+K(f.>'+f.V), . .), 
whence, after all reductions, the inverse function is 
{(A, . .Ja, i'X^, '/?, K )\/ (-"^5 • -X^'^ • -X^'^ "'XS’ ?)%/ (^5 • -X^? 0"}"’ 
and the remainder of the process of decomposition is effected without difficulty. 
Additioiv, 18 December, 1862. 
The formulae II. and II. (bis) each of them give the tangents of the conic 
(a, . . .\x, y, zf—^ at the ineunts of intersection with the line ^x-\-rly-\-'^'z=i). A very 
elegant formula for these ineunts themselves w'as communicated to me by Mr. Spottis- 
WOODE, and I have since found that the same or an equivalent formula is made use of 
by M. Aronhold in his recent valuable memoir, “ Ueber eine neue algebraische 
Behandlungsweise der Integrale irrational er Differentiale, »&c.”, Crelle, t. Ixii. 
pp. 95-145 (1862). The formula is as follows, viz. for the conic and line, 
(«, b, c, f, cj, hjx, y, zf=Q 
then 
x:y: z=. 
{J^-\-mrl+n^’) cyi) — ^\hl-\-bm-\-fn) + l \/^, 
: (Zl'-f m;?'+<') ^ +^'(«^+Am+^>?)— ^{gl-\-fm-\-cn)-^7ns/^, 
'■ {l^ -\-7nyl s/ <P ; 
