284 Mr. A. Cayley on the Integral f^m-r- \^ (x + a)(x+b)(x4-c)|l 

 we have in like manner 



±ii{k)±u[p)±i[[e)=o. 



And assuming a proper correspondence of the signs, the elimina- 

 tion of n(/}) gives 



t. e, if the points P, P' upon the conic x^-\-y^+z^=^0 are such 

 that their parameters 6, & satisfy this equation, the line PP' will 

 be constantly a tangent to the conic 



Kx" + y2 + 2^) + [ax^ + Jy2 + cz2) =0. 

 Whence also, if the parameters k, A/, A" of the conies 



y {a;^ + y'i + z^)+ax^ + by''-{-cz^. =0 



l(!'{x^ + y^-{-z^)+ax^ + by'' + cz^ = 



satisfy the equation 



Uk + Uk' + Uk"=0, 



there are an infinity of triangles inscribed in the conic 

 x'^-\-y'^ + z^ = 0, and the sides of which touch the last-mentioned 

 three conies respectively. 



Suppose 2Ilk=zIlK (an equation the algebraic form of which 

 has already been discussed), then 



ni9'-n6>=nw, 



6=cc gives 6'=k; or, observing that ^ = oo corresponds to a point 

 of intersection of the conies x'^ -]- 7/^ + z^ = 0, ax'^ + by^ + cz'^ = 0, 

 K is the parameter of the point in which a tangent to the conic 

 k (x^ + y^ + -5^) + ax"^ + by^ -\-cz'^ = at any one of its intersec- 

 tions with the conic x^ + y'^-^z^=0 meets the last-mentioned 

 conic. Moreover, the algebraical relation between 6, 6' and k 

 (where, as before remarked, k is a given function of k) is given 

 by a preceding formula, and is simpler than that between ^,^'andX:. 



The preceding investigations were, it is hardly necessary to 

 remark, suggested by a well-known memoir of the late illustrious 

 Jacobi, and contain, I think, the extension which he remarks it 

 would be interesting to make of the principles in such memoir 

 to a system of two conies. I propose reverting to the subject in 

 a memoir to be entitled " Researches on the Porism of the in- 

 and circumscribed triangle .'' 



2 Stone Buildings, Feb. 16, 1853. 



