Mr Rogers, On a Gaussian Series of Six Elements 259 



where e, = , , (- . 



7(7+1) 



But from (2), by changing /a, 7 into fii, y^, which does not 

 alter k, we have 



^here .. = (7 - «+ 1)(7- ^ + l)(7-^-+ 1) (/. + 1) 



(7 + l)(7 + 2) 

 These two results lead to 



H{a, ^,\, fx,, 7i) ^ 1 gi fr(ofi,^i,Xi,/X2,73) ^ 



Zr(a, /3, \, yu, 7) 1 - (>c-fx-iy-d'- H(a,,^„\„fJL„ry,) " 



The ratio of ^-functions on the right-hand side differs from 

 that on the left-hand side only in that a, /3, \, /x are increased 

 each by unity, and 7 by 2. This does not alter k — fi, so that we 

 have finally 



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



It is remarkable that if we change /x into y—fi, the coeffi- 

 cients e all become symmetrical in a, /3, X, /u., and « — /^ — 1 

 becomes ^(a + ^ + X + fx-l- 27), so that 



E{a, l3, X, 7-/^+1, 7+ 1) 

 f/^ (a, /3, X, 7 - ^, 7) 



is symmetrical in a, /S, X, yu^. 

 Hence 



Hja , ^, X, 7 -/,t, 7) ^ ^(g, /3, X, 7 + 1 -/x, 7 + 1) 

 H{a, l3,y-\, ix,y) H {a, /3, y -\-l-\, m, j + 1) '"^ ' 



where the first of these fractions is the same function of 7 as 

 the second is of 7+I. 



