348 REPORT— 1869. 



^(«,/3,7-l)-0(a,/3.y) 



=^^-''^'(fe^?^^^('^ + l'/5 + l'V + l) .... (3) 

 0(a,/3 + l,y + l)-0(a,/3,y) 



0(a,/3 + l,y)-^(a,/3,y) = 2%'j^^(a + l,/3 + l,y+l) . . (5) 



^(a + 1,/3, y)-^(a,/3,y)=9».r5-=|^^(a+l,^ + l,y + l) . . (6) 

 0(a + ],/3,y + l) 



=?''^Xr:E^^£^^(- + l,/3 + l,7+2) (7) 



= -5r-U-l=^!_!:^^(a,/3 + l,y + l) (8) 



^(a+l,/3-l,y)-^(a,/3,y) 



=2''^'^^^^(a + l,/3,y + l) , (9) 



These formiilge are applied by Heine to the expression of the fundamental 

 series in continued fractions. Thus from (4) we immediately obtain 



»(a,/3 + l,y + l) _ I 



0(a./5,r) i_g3-, (l-r)(l-g"-0 <p(a + l,li + l,y + 2) 



J- (l_2y)(l_jy+.)- ,^(a,/3 + l,y + l) 



Since 



0(a + l, |5 + 1, y + 2) _ < ^(/3 + l, g+l. y + 2) 

 0(a,/3 + l,y + l) 0(/3 + l,a,y + l) ' 



by the very nature of the series represented by the functions, we can repeat 

 this process and expand ^*-'''P+ ^7+ ) -j^ ^ continued fraction. The re- 



suit is 



^fg./g + l.y+l) _J_ajX a^ a^ a^ 

 ^(a,/3,y) l-_l_l_i_i_' 



where 



' ^ (l_2y+2)(l_2y + 3) ' ^ ^ (l_27+3)(l_jy+4) • 



When /3=0, (p(a,(3,y) becomes unity, and we obtain a continued fraction 

 for 0(1, a, y + 1), M'hich is immediately applicable to elliptic functions. 

 From the nature of the series we have at once- 



0(«+l,/3 + l,/3 + l)=.^(« + l,/3,/3). 



