262 Mr Rogers, On a Gaussian Series of Six Elements 



A continued fraction of the type 

 1 e^t e^t 

 1 + T+T+'" 



converges, if when reduced to the form 



1 1 ]_ 



1 + rfl+ C^2+ '"' 



either or both the series 



l+d2+di + ..., di + ds + d5+ ... 

 are divergent. 



That is l+^f!-i + ^^ + !lM!+., 



J- / J- 69, 62 ^4 



^ m 



y 1 



and ^ _+^+_riri_+... 



must be either or both divergent. 



In the first the ratio of the general term to the preceding is 



-^^^^ and in the second it is — ^^ . 

 62W 62714-1 



The first ratio is 



{a+n-l)(l3 + n -l)(X + n-l)(y- iii + n-l){y + 2n) 

 (7 - a + ?i) (7 - /S + ?i) (7 - X + ?i) (/u, + n) (7 + 2n - 2) 



= l--{l + 2(y+fi+l-a-^-X)] + 0(k], 

 n ' \n-/ 



while the second is 



(y - a + n) {y - ^ + n) (y -X+ n) (y + 2n + 1) 

 (a + n) (/9 + n) (X + n){y-fi + n) (7 + 2ii - 1) 



= l-^Jl-2(7 + ya+l-a-/3-X)| + 0Q. 



It is evident that one series converges and the other diverges 

 and therefore the continued fraction converges. 



In the Heinean case these ratios are approximately 



Q {a+P+\—y—ii—l} ^(y+;u.+l— a— A— m) 



one or other of which must be greater than 1. Hence one series 

 must diverge and the continued fraction converges. 



