on the lately proposed Logarithms of Unity. 285 
coincide with what is ordinaily understood by a” in all cases 
where the meaning of a®. has been fixed by usage, and to 
confer a reasonable analogical meaning on a* where usage has 
been silent. The definition of a* which I adopt is this: a* 
denotes in succession every function ¢ x of x, which, inde- 
pendently of x and y, fulfils the following conditions: 
or. oy = o(rt+y) (9.) 
aca (10.) 
This definition is, perhaps, the simplest that could be pro- 
posed, the most extensively applicable and the most accordant 
of any with other less fundamental properties of a” which have 
been observed to hold good within certain limits. 
The abstract contained in the Fourth Report of the British 
Association does not contain the reasoning by which I find 
JF (x f-'a) to be the general solution for a*, f§ being equal to 
cos 6 +4/—1 sin 6. My reasoning is as follows: 
I first proceed to show that if fx and fx denote two func- 
tions of x, each of which fulfils condition (9.), we shall have 
Jf 2 = f(x), c being some constant. 
Let f' 2 = fa, yx being some unknown function of «, 
the form of which is sought: then by the assumed property 
of f’, we shall have fpa. fly = fv (x+y); but by the 
same property of f,; we have fla. fly =f(ba+ vy). 
Hence fy (x# +y) =f(va+ vy). 
Now, if ¥ (x + y) differ from Pa + Wy, let 64+ ) (7+ y) 
= x+y; then we shall have f{6+¥(x+y)} =f(ve 
+vy) =f (c+y); but by the property of f f {6+ (x+y) 
any + y); hence fo. fp (e+ y) =f (x + y); hence 
il. 
Hence the general equation to find Wx is 
b+0(r7+y)=vetby, (11.) 
§ being some quantity such that f9 = 1. 
Let Pr—b=We (12.) 
By this substitution we obtain from (11.) 
Vety=Vordy (13.) 
I consider equation (13.) the purely algebraic part of the best 
definition that can be given of what is meant by multiplication 
in its extended sense, since that definition is based on the 
most characteristic formal property of multiplication in arith- 
metic, the science suggestive of symbolic rules, In my view 
of algebra, if we presuppose arithmetic in general, and alge- 
braic addition, we may at once, on having obtained equation 
