166 
equation derived from (5) by repeating once more the process which 
leads to the latter equation. So we put 
k= fa de . sje, | ain ee oe ae RA 
0 
Then, again, A(t) is continuous and differentiable in (0,1) and 
we have 
WAG OLY osha ncn «a. etree ae (8) 
The principal point, however, is that the latter equation is also 
valid at t=0. Thus the derivative of A(t) is a Limited function in 
the closed interval (0,1). Further, observing that 
lam [A (i) >t] =Af' (0) =g (0) = 0, 
t= 
we find on integrating by parts, for Rx) > R(c) 
1 1 
fo (Deel dt = h (1) — (w—c—1) fi Qt dt ao) 
0 0 
and hence 
1 
a (w) = g (1) — (w—c) h (1) + (@—e) (w@ —e—l) JA (t) Be? dt. (10) 
0 
3. The preceding statements are valid independently of any 
further hypothesis as to the character of f(t. Now, suppose that 
a(«) becomes zero for the arithmetical progression of values 
=d U, (OLED DUA DD ae eel 
Choosing the number c in the preceding equations equal to & we 
find g(1)—0 and the integral in the right-hand member of (5) 
vanishes for 
a=G§+Il+4, (SSO i eis nn te en 
From this it follows that A(1)=0, and, in connection with the 
latter result, from (40) 
1 
frova=o, (u), bende Ades a ES 
0 
Now we saw that the derivative of h(t) is limited. According to 
a well-known proposition A(t) can therefore be expanded in a series 
of Fourier. We have 
h() = Sn (an cos 2a nt + by, sinQant) . . . . (14) 
0 
