284 Mr. Graves’s Reply to Prof. De Morgan’s Remarks 
of real values, which are successively equal to 1, and to the 
. . —din*®_. . . 
ascending integral powers of e, 4’ o° with their reciprocals. 
The expression e,~4'™’ is equivalent to cos (/ —1 477°) 
+7 —1 sin(/—1 422°). In these investigations there is 
often much convenience in thus putting imaginary quantities 
after sin and cos, and the want of familiarity with this practice: 
has, perhaps, been an obstacle to the full comprehension of 
the new theory. 
There is another objection started by Professor De Morgan 
but not pursued. He says that after supposing a = /4, 
where f§ = cos 6 + 7 —1 sin 6, I find a = f(xf-a), but 
that I should have obtained precisely the same expression, if 
I had assumed 6 = e°*, c being any quantity whatever. By 
e°*, I here understand the Professor to mean ‘hat particular 
cé site c 6 (c 6)* (c4)° 
value of e-~ which is equal to 1+ — + eo ay 1.2.3 
1 
and which I should denote by ey *. On this understanding 
I fully agree with his statement, but not with the inference 
which I suppose him to draw from it. I suppose him tacitly 
to infer, without minute examination, that when /@ = cos 6 
+ eee, 
+ s/—1 siné, or &”“—14 the expression f(2f-!a) cannot 
have so general a meaning as it would have if the meaning of 
f$ were generalized in the mode he mentions. ‘This is an in- 
ference which, if the functional definition of a* adopted by 
me were admitted to be proper, would at most only charge 
my solution with not being sufficiently general. I acknow- 
ledge, indeed, that the impropriety of that definition would 
be evinced, if it fairly led to a meaning of a? which subverted 
any established exponential theorems more fundamental than 
those which that definition embodies, or if by the inconvenient 
generality of the values it would comprise, the expediency of 
a further limiting equation of condition were pointed out. 
The supposed inference, however, as above stated, is not cor- 
rect, for it happens that even if /@ were assumed equal to 
2 3 
oh Leh 
1 1.2 1.2.3 
be unaffected by any variation of c, since it would indicate 
operations which would eliminate ¢ from the result. That re- 
sult would in consequence be identical with mine, and if ex- 
amined with a little attention, would be perceived entirely to 
1+ 
+..., the expression f(x f~'a) would 
