1 



258 Mr Rogers, On a Gaussian Series of Six Elements 



To prove this write the equation in the form 



where the ^'s, Eb and C's are independent of ^, it being observed 

 that, m the first and third series in (2), k becomes « + 1. 

 The left-hand side is 



Hence "--^ (..'-^.) W-<^') + -• 



Co = ^2 - 2/CyU, + 5j — ^ J 



7 + 1 ry 



= ,7(;y^) (t - «) (7 - /3) (7 - X), 



while r;„ = 5„+, + 5,, {(^ - f,y _ (« + ,,).} _ ^^^^^, 



^^^ ^-+^ = ^-^TTV ^- and A.+1 = ^+^^»+^W/ i.^ 

 (^ + 1)7 "+^ ('^+l)7.+i '^' 



so that §-^ = g>mgn+iX»4-i^n _ /2/. _ ^ 4. „ w, °^^'"- 



^(7-«)(7-^)(7-X) 



tTt+T+I) ^'^^ (3), 



as may be easily verified, thereby giving the right-hand side of ^9^ 

 and establishing the relation. ^ ^^ 



We may now obtain a continued fraction of the form 

 1 e^x e^x CsX 



i-r^r=n: w, 



