Crofton — Solution of Certain Functional Equations. 21 
where, moreover, the constants Ci, C^, may be of such forms as 
t/itan -— , &c. 
o 
2. To find a solution of 
hence, as before, 
log w . = log a.y \ 
log a 
3. If (j>{mx) = a<fi{x) + J^j?'', 
m^^y — ay = hx^; 
.-. y = 1 (m^^ - = + Cx 
mf - a 
4. Let us take Laplace's equation — 
^{rnx + w) = a<j>{x). 
We may express this as 
<f>{my + c) = a(f>{y + c), 
where y = x - c. c = , ^ . 
is'ow, let ^{y)^^{y ^ o), 
and by (2), above, as ^{jny) = a\l;{y\ 
il/{y)= Cy^ogm-^ 
.-. <jE)(y + c) = = C( ^ - — m)^°^'^' 
In the Proceedings of the London Mathematical Society, vol. xii., 
I have shown that iiz, x are related by the condition 
i(/{z) ^ 1 + lA(^), 
* This might have been found from (1) ; as if we put 
<J> {mx) = ;|/ (log X + log m) = a\p (log x) ; 
log;tr log a 
^ (log x) = C^j = Ot; 
