282 Mr. Graves’s Repiy to Prof. De Morgan’s Remarks 
arithmetical value of a’, I must say that I should not approve 
such a restriction. It would, if the proposed theorems be cor- 
rect, arbitrarily exclude from the xame of logarithms orders of 
functions which enjoy the same fundamental and characteristic 
properties as those that are favoured with that name. It would 
also in some cases be difficult of application. The expression 
= a = 
(m—/ —1%)? has two values. If we adopt the restricted 
definition, of which of those values shall we be allowed to call 
1 a logarithm with reference to the base m— 7 —1n? On 
the other hand, there are cases where a” has an infinite num- 
ber of real and positive values. But I proceed to prove that 
Qin 
1 is among the values of e 2¢m— V —1 from postulates which 
I believe will be conceded by Professor De Morgan. 
For all values of x, real or imaginary, by cos z, I denote 
a x x 
1 eh een Asse a ee 
and by sin x I denote 
wv a 2 a’ 
TRILL SST ML SAG LLSAse Cee 
I assume that 
j 2 qa? 
Rebige Tongegnit ages? Pe: 
Athi y* y 
sles fig Wye tout 
(vty) , (waty)? , (e@+y)* 3 
it 1 Tantdes T 7.2.3 vip (3.) 
that the complete and correct equation for e” is 
aneee x x a 
Ante (14 ig. Ce aes (- 
and that, if w denote in succession O and all integers positive 
* and negative, the values of 1° are properly represented by the 
formula 
cos (2wam)+* 7 —1 sin(2wez). (5.) 
* I have found that upon the whole it tends to superior clearness of 
notation to place ./—1 foremost.where it occurs as a factor. 
