444 Proceedings of Royal Society of Edinburgh. [sess. 
{ „ ! 4 FTi DW • 9 - Z pr(b - n+r) { n-r\ B 
{■' l 
i n ) 
1 + 
j a + 1 
1 n 
b t [T]T 
n) 
a + l ( I 
n ( ] 
i a + 2 ) 
1 n j 
n i 
f b +■ 1 ( 
1 m 1 
x [2] 
+ . 
- i a 
I n 
1 + 
i"‘[ 
1 a -r 1 
) n 
m 
: a : 3 } 
14 
< 
1 6+1 1 
1 « ) 
‘xPW 
in w 
W 
: Pl 2] 
ol! + • • 
] 
( 8 ) 
When p = 1 the right side of the equation identically vanishes, 
being of the form f{x) -/( x ?) . 
(4.) 
i « i 
I n j 
The function yjT\ Can *’ C trans f° rme( l into F([«] [/?] [y]) by 
1 n \ 
any of the four following substitutions : 
(1) 
y - a - (3 - 1 
7-0-1 
-0 
(V 
7-1 
y-p-i 
a 
(3) 
y-a-fi-l 
7 - a — 1 
— a 
(4) 
7-1 
y — a — 1 
/ 3 
and since 
a l J « i 
n-r ) j n I p n - 1 • p 71-1 - 1 • p n ~ 2 - 1 ... . p n ~ r + 1 - 1 
( b I ( b I J, a ~ n + l - 1 . pa- »■+* — 1 pCt - n+r _ | 
i n J \n] 
Substitution (1) transforms this expression into 
