260 Mr Rogers, On a Gaussian Series of Six Elements 

 § 3. From § 2 we have 



1 e, e. 



.(1). 



i-iK-fi-iy-e^-i- 



Now change a, /3, \, /n, y respectively into 



7 - a, 7 - /3, 7 - X, 7 - /i - 1, ry _ 1. 



It will then be seen that « — ^— 1 becomes -(/c — /* — 1), 

 K becomes — (« - 7— 1), e^ becomes e^, and generally en becomes 



We have in (1) then the same continued fraction as in § 2 (4), 

 and hence 



H{a, ^,\,fi,, 7i) 

 IT {a, /3, \ fx, 7) 



= (^-/^-l)'-^' H (71 - g, 7, - /3, 7i - X^ , 7 - At, 7i) 

 (ac-7-1)2-^2 H{y-a,y-/3,y-\,y-fj,,y) ' 



Q^H{y-a.,y-^,y-\,y-^^y) 

 H{a,^,\,fM,y) 



= (^-/^-l)'-^- H{y,-a,y^-^, 7i-\, 7-^, 7, ) 



(/c-7-1)-^-^^ i/ («, /3, X, /x,, 7,) ^^''• 



Let X = 0, then 



H(y-a,y-^,y-/ji) 



or, changing a, /3, /x into y ~a, y - (3, y - ix, 



The series ^ (a, /9, yt*) has been investigated by Saalschiitz, 

 Zeitschrift f. Math, xxxv., for the case when a is a negative integer, 

 and generally by Dougall, Proc. Edinburgh Math. Soc. xxv.,§ 12 (20). 

 For special cases of the continued fraction § 2 (5), see Proc. 

 London Math. Soc. series 2, vol. 4, pp. 83, 394. 



§ 4. Heinean forms. 



The results of the foregoing sections have their counterparts, 

 when every factor in the terms in H, are replaced by correspond- 

 ing sines or hyperbolic sines. 



