340 Mr. A. Cayley un the Porisni of 



and for n = 4, 6, 8, &c. respectively, the conditions are 



D I =0, 



= 0, &c. 



Theorem. In the case where the conies are replaced by the 

 two circles x^ + y^—W-=-0, (^ — fl)^ + ?/-— r^=0, thenthe discri- 

 minant, the square root of which gives the series 



A + B| + Cp + Df+E|4 + &c., 

 is 



Write for a moment 



A+B^ + Cf + Df + E|'» + &c.= v/(i + «^)(l + 6?)(l+ca 



then 



A = l 



2B=fl + 6 + c 



-8C = «2 + 6'^+c2-2Z»c-2ca-2ff6 



\QJ) = (fi-\-¥ + c^-a^{b + c)-b'^{c + a)-c\a-\-b)-it'iabc 



-128E = 5«'» + 5Z»4 + 5c'' 



-4«3(i + c) -453(c + «) -4c3(fl + 6) 



+ 4ia%c + W-ca + '^c'^ab 

 -26V-2cV-2«262 



&c. 

 To adapt these to the case of the two circles, we have to write 



and therefore 



c=l 



values which after some reductions give 

 A = l 



?-22B = 2;-2 + Il2-«2 



?-6l6D = (R2_«2) I (R2 _fl2)2_3r2(R2 + a^) } 



-?-n28E = 5(R2-0'' 



-8(R2-a2]2(R2^2?-2)r2 



