1902-3.] On the Series y — 1 + F((a] [5] [y]) . ^ + , etc. 445 
. m/i+lW + 2\ r/8 + r-ll 
[y - a] [y — a + 1 ] [y - a, + 2] . . . . [y — a + r — 1] 
and the differential equation takes the form 
F(w m m) { 2( - [ |Sfe 
.. [/3 + r- 1] a? [r] 
. . . [y - a + r- 1] [r]\ 
,.,[ 7 +r-l] [r]! 
[y-a]ty-a+ 1J . . 
D« -y\ - V ( - 1 '•jpKr-ffi- 9 
— + !]• 
[y] [7 + 1] • • 
3™ 
A \ 1 + F([a — 1] [/3] [7]) * [jjj 
F([- - 1 ] [jSf] M) - F([- - 13 D3] [y + 
xm 
- A -J 1 + F([a- 1] [/?] [y]) • + 
The second substitution gives 
(9) 
-a)+ 
r.r-l["a] \a 
~ LJL 
r-r - 1 [a] [a - 1] .... . [a — r + 1] 
[y - a] [y - a + 1 ] . . . [y - a + 7 
1] . . . [a - r + 1] X^ r 
' py 
[y — a — /3] . .. [y-a-/? + r-l] 
1] ' [r] 
D(-+iV 
xW ) 
FI"' } 
= A 'j 1 + F([tt] [/? + 1] [y+ 1]) • j-jj, 
+ F([«]DS + l][y+ 1] • F([«] D8+ 1] [y + 2 ] • g + . . . } 
— A | 1 + F([a] [/3 + 1] [y + 1]) | (10) 
Substitutions (3) and (4) simply interchange a and ft in the two 
equations given above. 
These differential equations have a solution 
™[1] r [2] 
?/ =l + F([a][fl[y]) . [ - I J, + F([a]M[y])F([a]M[y+l].p ]! + 
If we make any of the following substitutions, 
