140 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
from which he concludes that 
A h — A k contains 2« as a factor. 
Thenceforward he specialises. Taking the case where n is prime, and 
where, therefore, the n roots are 1 , 6 , 0 2 , . . . , Q n ~ x , he denotes the product 
of all Spottiswoode’s factors except 
by 
a : + a 2 + ... + a n and FVq + 0 n l a 2 + ... + 6a % 
l \ + b 2 0 + 6 S 0 2 b n 6 n ~ l 
and establishes the result 
C = 2a. 
nZab - (2a)’ 
72—1 
where 2 ah stands for a-fi 1 + a 2 b 2 + . . . + a n b n . The expressing of the b’s 
in terms of the <x’s is not attempted. Further specialisation affects the <x’s, 
the investigation (pp. 377-380) becoming more intimately associated with 
the theory of integers. 
Glaisher, J. W. L. (1872). 
[On functions with recurring derivatives. Proceedings London Math. 
Soc., iv. pp. 113-116.] 
The functions referred to are 
1 H 1 + ... 
n\ (2ra)! 
„2n+l 
X + 
(ra + 1)! (2w+l) 
+ . 
1 1 yrg 3w 1 
+ + 7TT" — . + . • . . 
(w — 1) ! (2n — 1) ! (3n-l) 
which are readily seen to satisfy the differential equation 
d n u 
— — — u. 
dx n 
the derivate of the first function being the last, the derivate of the second 
being the first, and so on. Denoting them by </> 0 , <p x , 0 2 , . . . , , Glaisher 
of course obtains at once 
$0 + + ^2 + 
• • • + < t > n - 1 — & 
00 + + #102 + • • • 
+ # i _ 1 0 «-i = e° lX 
00 + #»-101 + #»-102 + • • • 
