282 Mr. A. Cayley on the Geometrical Representation of 



Suppose that the tangent meets the conic a7^ + y' + r^=0 (which 

 is of course the conic corresponding to zi; = x ) in the points P, P', 

 and let 6, x be the parameters of the point P, and 6', x the 

 parameters of the point P', i. e. (repeating the definition of the 

 terms) let the tangent at P of the conic cc'^-{-y^-\-z'^ = Ohe also 

 touched by the conic 6 [x^ + y' + z^) + ax^ + hif + cz^ =0, and 

 similarly for 6\ The coordinates of the point P are given by 



the equations 



X'.y \z^ s/h — c s/a-^Q \ \/c—a Vh-\-Q \ Va — b \^c-{-d', 



and substituting these values in the equation of the line PP', we 

 have 

 (b — c) \^a + k \^a+p \/aT6+ (c—a) \/b-\-k \^b-\-p \^b-\-d 



+ {a—b) \/cTk\^c-\-p \/c-\-6 = (*), 



an equation connecting the quantities p, 0. To rationalize this 

 equation, write 



\^(a-hk){a+p){a-^6):=X+/jLa 



\/{b-\-k){b-^p){b + e)=X-\-fjLb 

 \/(c-hk){c-^p)(c-h6)=\ + f^c, 



values which evidently satisfy the equation in question. Squaring 

 these equations, we have equations from which X^, Xfi, fx^ may 

 be linearly determined ; and making the necessary reductions, 

 we find 



\^=iabc-{-kpe 



-'2\/jb=bc+ca + ab—(p6 + kp-\-kd) 

 fju^=a + b + c-]-k-\-p + 6. 

 Or, eliminating \, yit, 



{bc-hca^ab-(pe + kp-^ke)}^ 

 —4:{a-\-b + c + k+p + 6)(abc + kp6)=0 (*) 

 which is the rational form of the former equation marked (*). 

 It is clear from the symmetry of the formula, that the same 

 equation would have been obtained by the elimination of L, M 

 from the equations 



\/(k + a){k-^b)(k + c) = h + Mk, 



\^{p + a)(p + b){p-{-c) = lj + Mp, 

 \/(d + a)(e + b){e + d) = L + M(9. 



And it follows from AbePs theorem (but the result may be 

 verified by means of Euler's fundamental integral in the theory 

 of elliptic fimctions), that if 



dx 



Ux=zf 



00 \^{x-\-a)(x-}-b){x+c)' 



