96 
Proceedings of the Royal Society 
More generally, 
and 
P n _ ( df Pn-i/ 
whence 
\dpj p n ~ v ’ 
P = (1_Y p n(±Y 
” v \dpj “ \dpj p n ~ v 
Changing n - v to ra, this becomes 
— m p 
m 
m i 
which, when ra = 0, agrees with (7). Here n may have any posi- 
tive integral value not less than m. When we write n~m we have 
merely a truism. If we put n = m-f 1, we arrive at the same result 
as we should have obtained directly from the first forms of the 
equations (5) and (6). All these series satisfy differential equations 
of the form 
x 
cPy 
dx 2 
= *• 
Corresponding properties are easily proved for the series forming 
the co-efficients of the various powers of x in the expansions of 
expressions like 
€^ W + ^, € ^ + ^\ &C, &C. 
It is easily seen that what has been called P 0 above is the 
infinite series 
p o = ^ + 12^32 + • • • = f(p) • • ( 8 )> 
and that quite generally if 
n 
m 
_l+_P _j__£ 
I "I MiOm i 
pm 1 pm^pm 
&C. 
we have 
“-(4)V(4 W '" 
n 
m 
whatever positive integer be represented by n. Of this the simplest 
case is U 1 = € p > where of course 
