Mr Rogers, On a Gaussian Series of Six Elements 261 



If the fundamental series be taken as 



sinh ma sinh mB sinh m\ sinh mu, 

 sinh ra sinh 7717 sinh m {k — 6) sinh m (k + 6) 



then, if e^^m ^ ^^ ^Ijq factors in ijj take the form 1 — g", etc., there 

 being an extra factor 



exp m(a + /S + \ + /i-l-7- 2/c), = e"'"^ = 5. 



Moreover 



t>^+i _ (1 - q-+^){l - qi^^'') (1 - 9^+'0 (1 - (/^+^0 

 ?;„ (1 - ^«+0 (1 - 9T+'*) (1 - 9«+«-») (1 - ^-c+^+s) ^' 



so that the series is obviously convergent if | g- 1 < 1, and no indices 

 of powers of q are zero. 



The formula corresponding to § 2 (1) is readily obtained, 

 giving 



_ sinh mcL sinh myS sinh rti'^ sinh (7 — /i) 

 ^ sinh rri'y sinh wi (7 + 1) 



That corresponding to § 2 (2) is more difficult, but it depends 

 on the fact that, by employing the somewhat intricate identity 



sinh m (a + n + 1) sinh m{Q + n-\-\) sinh m {\ + n -\- \) 

 sinh m (w + 1) sinh m (7 + w + 1) 



_ sinh ma sinh m/3 sinh mX, 

 sinh wi (/I + 1) sinh my 



— sinh m (a + yS + X, — 7 + w + 1) 



_ sinh m (7 — a) sinh m (7 — /3) sinh m (7 — X ) 

 sinh /?<7 sinh TO (7 + ?i + 1) 



we get, corresponding to -jy* in § 2 (3) the value 



sinh m (/x + w) sinh 7?i (7 — a) sin h m (7 — ;5) sinh m\'y — X.) 

 sinh m7 sinh m (7 + 71 + 1) 



This leads to a relation corresponding to § 2 (5), where 



- = sinh^ m (k— a—l) — sinh- md, 



a; 



and the e's are altered to corresponding hyperbolic functions as 

 explained above. 



§ 5. Convergence of the continued fraction in § 2 (5), when x 

 is negative and the e's are all positive. 



