442 Proceedings of Royal Society of Edinburgh. [sess. 
So also if we operate with 
y K6 .„ +r) ( b !^ D(1 . +1) 
\n~r j [rj! 
on x^ we obtain 
r(b - n + r ] 
XP^ 
r) \ n b - r } ' jy[r 1] . . . [m-r\x W-[>-+i] 
0 
which may be written 
[ m ] ^pr(b-n+r) j ^ | | m “ * | 
which is /jjjp[i»-i] [ w ] . | ^ | 
If y = A^M + A 2 «[ 7W 2 l + A 3 « [m 3l + . . . . 
we shall have 
™[r] 
b I ^ [r] n 
2 ^- B+r) {n-r)w m) -y- ±p-^ \ n : r } 
0 0 
- Ajfmj] | !l + n h~ 1 J.a;P[mi-i]-A 2 [m 2 ]| b + v h~ 1 J. j-Mwa-U (5) 
It is not possible to choose values of m 19 m 2 ... . 
•A-l ? -^-2 • • • • 
to make the right side of equation (5) identically zero ; but if we 
choose 
m 1 = m 2 — 1 
m 2 = m 3 - 1 
m r = m r+ 1 — 1 
Ar+iK+i] j b + mr ^ " 1 } = A, { “ + "* } ... .(6) 
the series on the right become 
A i a + m A a/mj + A i |“ + ffl i +1 | «^+« + . . . . 
1 { w j 2 ( n I 
— A 2 [m 1 + 1] [ | — Ag[/7' 1 + 2 ^ "'l ^ | xA“.+i] - . . 
