on the lately proposed Logarithms of Unity. 287 
I have perused with interest Professor De Morgan’s ob- 
servations on the impossibility of ever proving that we have 
arrived at the most general solution of a functional equation. 
It seems to me, however, that our only limit to the meaning 
of the general symbols employed in the solution of such equa~ 
tions is the necessity of their compliance with certain formal 
conditions, which must by tacit or express convention be 
considered elementary and definitional. Now, I think, we 
may sometimes show that given functional equations can be 
solved by the solution of certain others expressing such ele- 
mentary conditions, and by such solution only. When we 
can show this, we are at liberty, in my opinion, to substitute 
the symbols which the latter equations define, and to pro- 
nounce ourselves in possession of the most general solution. 
This subject deserves further consideration. 
There is an error in the abstract of my last memoir. I 
there appear in substance to define cosine and sine to mean 
respectively such functions ¢ and as simultaneously fulfil 
thefollowing conditions : 
guoy —vabvy = o(e@+y) (16.) 
oubyt+veoy=vurty) (17) 
(¢2P + (var)? = 1 (18.) 
These conditions do not constitute a sufficiently limited de- 
finition to coincide with the ordinary acceptation of cosine and 
sine. The general solution of (16.) and (17.) gives ¢a 
j ays if pate » 
a Sex) cake ze) and va = fies) Le 2), where S§ 
means, as before, cos§ + / —1 sin 4, cos and sin @ being 
defined by equations (1.) and (2.).. The third condition (18.) 
only requires that c’ should be equal to —<, and so only 
limits $v to cos (ca) and Y2 to sin(cx). If instead of the 
third condition we were to substitute the following, viz. 
dy: 
1 by ele 0) ote 5 
, we should have a good definition coinciding with 
dua Oo 
(1.) and (2.). 
I am anxious to embrace the present opportunity of cor- 
recting a former involuntary misrepresentation with respect 
toProfessor Ohm. I find that his logarithmic formule are 
not only coincident in principle with mine, but coextensive in 
their applicability to imaginary as well as real quantities. 
On some future occasion, Gentlemen, I shall be happy, 
with your permission, to communicate my investigations re- 
lating to the limits of the possibility of finding a base , such 
or = 
