298 
[686 
686. 
ON A FUNCTIONAL EQUATION. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xv. (1878), pp. 315— 
325; Proceedings of the London Mathematical Society, vol. ix. (1878), p. 29.] 
I was led by a hydrodynamical 
ax + b 
viz. writing for shortness x 1 = 
±d 
problem to consider a certain functional equation; 
this is 
(f)X — (f)X 1 = (x — x x ) 
Ax + B 
Gx + D ' 
I find by a direct process, which I will afterwards explain, the solution 
_ A f{(a - d) 2 + 46c] (AD - BG) f“ sin ft sin yt dt _ 
(px — q x + (7 (d(7 _ J 0 s j n ^ sinh 7ri 
where £ is a constant, but £, ?? are complicated logarithmic functions of x (£, rj, K 
depend also on the quantities a, b, c, d, G, D); sinh 7rt denotes as usual the hyperbolic 
sine, \ (e* 1 — e~ nt ). 
The values of £, 77, £ are given by the formulae 
1 _ a" + d 2 + Zbc 
X + \~ ad-bc ’ 
a = a« + 6, b = — dx + b, 
c = cx +d, d = cx — a, 
W=Ga + Dc, 
Z=Cb + Dd, 
P = Xc -4- Xd, 
S = — c - d,