Mr. Jerrard on the Occurrence of the Form g . 345 



From (7.) and (8.) C^ - B^ (6^ - c^f - A^ K " <^Y ^^ 



= AB^3/(-J+ |) x{a^-c'W--c') (10.) 



From (9.) and (10.) A B {a^-h') {a^-c^) [b^-c^)^y^ 



^ a^c''- (a'-c*) (62-c2) C^c'^z'^ 

 -(a\{b^-c^f-b''{a^-c^){b^-c^)) Wy" 

 -.(fl«. [a^^c^f-a" (fl^-c^) (i^-c^)) k^x" 



r^a^c^-c^z'-c^y^-a^x^ (11.) 



From (11. )> interchanging («, ^, f) with (J, 3^, ^r) we have 

 A B (6^-fl^) (62 -c^) (a2_c2) xy i-= b^c^-c^z'-c^ x'-b'^y^ 



(12.) 



Finally, from (11.) and (12.) we have 



(a^ c^ - ^2:=^, ^2 __ ^9 ^ ^rqry-qr^) 



X {b^c^-W^^^.y^-c^a;^ ^y^ + s;^) 



= («2_c^)(Z,2_^9)^2y 



i. e. [x" + «/^ + 2^) (a^ a;2 •\-b''y'^ + c" %^) 

 —a^ (b^ + c«) + ^2-Z>^ (a2 + c2) /— c^ (6^ + «') ^^ 



The = " required. 

 University College, March 5, 1838. 



LVIII. On the Occurrence of the form ^ in passing from ge- 

 neral to particular Values of certain Algebraic Functions. 

 By G. B. Jerrard, Esq,^ 



SUPPOSE that we have two equations for determining or, 



ar*»+A^*^^+ Bx"^-^ + V = (1.) 



^ = P + Q^ + R^2 + Lo;^ (2.) 



so that X- M3/>P>Q>R>-L) . 



so that x^ 4,(3^,p,Q,R, ... D' 



A, B, C, ... V being constant quantities involved in the mean- 

 ing of the functions <^ and v[/. 



Let us further suppose that P, Q, R, ... L are in such a 

 manner dependent on certain equations of condition that we 

 are able to foresee that the expression for x will be deter- 



* Communicated by the Author. 



