228 Mr Mathews , A representation of the exponential 
A representation of the exponential function as an infinite 
product. By G. B. Mathews, M.A., F.R.S., St John’s College. 
[Received 23 April, 1907.] 
Let z be a quantity of which the absolute value is less than 
unity, and let us assume that constants k 1} k 2 , etc. can be found 
such that 
z = k x log (1 + z) + L log (1 + z 2 ) + ... 
= 2& w log(l + z n ). 
( 1 ) 
Expanding the logarithms in power-series, and equating 
coefficients, we find that 
h= L 
while for all higher values of n 
k n + 2 ^7 — ks — 0 
with 88' =n, 8<n, and each proper integral factor of n supplying 
one term to the sum. 
Hence nk n = S (— l) s 'S&s = %(— 1 ) n/s 8k s , (2) 
by means of which the constants may be calculated. Experiment 
suggests the truth of the following statements, which will be 
confirmed by an inductive argument. 
(i) If n = 2 r 
i.-i 
K' il — 2 * 
(ii) If n = p 1 p 2 ... p fl) k n = 
(- 1 )' 
p lt p 2 , ... p^ being different odd primes. 
(— 1 
(iii) If n = 2 m p 1 p 2 ... p k n = . 
(iv) If n has an odd square factor, k n — 0. 
Cases (i) and (ii) are easily proved : thus by (2), if n = 2 m , 
nk n = X8k s = %%8' +1 (8' = 2, 4, . . . 2 m ~') 
= (2 m ~ 1 -l) + l =±n, 
assuming the formula to hold for lower values of ni : and since 
& 2 = 2 > tl ie statement (i) is correct. 
Again, if n=p x p 2 ... p^, every divisor of n is a product of 
different odd primes, and if h is the number of primes which 
divide 8/ t , any proper divisor of n, we assume that the formula (ii) 
holds for it, and thus obtain from (2) 
