(s.) 



566 Mr. G. B. Jerrard's Re/ieciions on the Resolution 



,4»^^^_ + ,(4 + 1)7.0^ ^_^ > 



the successive coefficients of 0„^;i being 1^", »5"+^', ,5n + 2A, 



,5n+3A^ ,5«+4A^ or (,A)0^ (,A)i^ (,A)2^ (,A)3^ ^,A)4_ 



And if we reflect that the system (0.) will remain unaltered 

 if, substituting .«0i, *2« 0^, ,3« q^^ ,4« q^ instead of ©i, ©2' 

 03J 04 respectively, we make suitable substitutions among ^r^, 

 ocpfl ccy, x^, Xi ; and that 0^, or r^_^v\ must be such as to ad- 

 mit of being equated either to r'^_n{p t)^ or to r"4_„(p^ u)'^ ; we 

 shall have little difficulty in perceiving that 



I e/ - i^ (^, + .^x^ + ^^x^ + .-^, + .-^/ 1 = 0. 



34. We are thus permitted to ussume 



0? = 4(^« + '''^^^ + "^^^ + '''^^j + '"^.n 1 



e'„^ eV being respectively equal to 7\_n [ptf, ^"4 _«{/>'*«)"• 



35. It follows therefore that 



0C = e"4„^ ....... (u.) 



Whence we deduce 



0\s = 0V. 0V = 0V» 



observing that 0^5a-|-i = Qb' 



Section VII. 



36. Let us now consider 0„ in relation to the different 

 values of P. To indicate that the expression which may take 

 the place of 0^ is a function of P/(ab)(ed) .. j ^ annex to 0„ 

 the index of P» Thus 



0«,/(ab)(cd)., 



