342 Mr. A. Cayley on the Porism of < 



from Jacobi's memoir, to be presently refen-ecl to). Fuss puts 

 R-f-a=j9, R — fl = 5', and he finds the equation 



rY-p\r' + q^)~-V ^' 



which, he remarks, is satisfied by r= —p and ?•= , and that 



•' ^ p + q 



consequently the rationalized equation will divide hy p + r and 



pq — r{p-\-q) ; and he finds, after the division, 



p'^q^+p''q'^{p + q)r-pq{p-\-qfr^-{p + q)[p-qYr^ = 0, 



which, restoring for p, q their values R + «, R — a, is the very 

 equation above found. 



The form given by Steiuer [Crelle, vol. ii. p. 289) is 



^(R_a) = (R+fl) x/(R_r+«)(R-?— fl) + (R+fl) v/(R-r-a)2R, 

 which, putting ;>, q instead of R + a, R — a, is 



qr=p \^{p-r){q-r)+p \^{q-r){q+p). 

 And Jacobi has shown in his memoir, " Anwendung der elhp- 

 tischen Transcendenten," &c., Crelle, vol. iii. p. 376, that the 

 rationalized equation divides (like that of Fuss) by the factor 

 pY— (7J + 5)?', and becomes by that means identical with the 

 rational equation given by Fuss. 



In the case of two concentric circles a = 0, and putting for 



greater simplicity —^ =M, wc have 



A + B^ + CP + DP + E^'» + &c. =(1 + 1) Vr+M|- 

 This is, in fact, the very formula which corresponds to the 

 general case of two conies having double contact. For suppose 

 that the polygon is inscribed in the conic U = 0, and circum- 

 scribed about the conic U + P^ = 0, we have then to find the dis- 

 criminant of ^U + U + P2, i. e. of (1 + |)U + P^ Let K be the 

 discriminant of U, and let F be what the polar reciprocal of U 

 becomes when the variables are replaced by the coefficients of P, 

 or, what is the same thing, let — F be the determinant obtained 

 by bordering K (considered as a matrix) with the coefficients of P. 

 The discrinunant of (1 + ?)U + P^ is (1 + ^)=^K + (1 + ^)^F, i. e. 

 (l+|)^{K(l + ^) + F} = (K + F)(l + mi + M^), 



where M = p — rs ; or, what is the same thing, IM is the dis- 



crimiuant of U divided by the discriminant of U + P^. And M 

 having this meaning, the condition of there being inscribed in 

 the conic U = an infinity of /t-gons circumscribed about the 



eonic U + P^ = 0, is found by means of the series 



A + Bf-|-C|« + Df ■fE^^^.&c. =(1+1) x/lTW 



