1248 
Thus we have established the result, that every function / (2) 
which satisfies the conditions of Drirrcurer for all values between 
O and oo may be expanded in a series of the form 
f(s) = 4% + 4, p, (*) + 4, p, (4) +... 0 Sr oo vald) 
where 
Ay =(—s pn (a) da 
0 
It is to be remarked that the values f(¢-+0) and f(c—0) being 
different, the second member reduces to } | f(c + 0) + f(c + 0)]. 
4. We now proceed to give two interesting examples of this 
expansion and to show the value of this expansion for the problem 
of the momenta. 
] 
l+ze 
As a first example suppose it is required to express f(x) = 
in a series of ABEL’s functions Pp (2). 
Evidently this function satisfies the conditions of Drirrcurer from 
2 = 0 to = om, thus 
1 
Teas au a sE a, Pz («) “lee 3° 
where 
ys Pn (a) da 
Wij == | El me 5 =* 
0 
Now the following relation holds between successive functions g: 
(n +- 1) pr (a) = (An + 1 — 0) pj (0) — rn Pii (a) <A e(D) 
)—0 
Multiplying this by da, and integrating between O and oo 
l+a 
we obtain 
ee 
(xn + lam = (2n + 1) a, — NAn—i cae Pn (a) da’ 
lea 
0 
But, as 
a 1 
== 
lta la 
we have 
i minn 
P a= fe “% pPnla) ad — ay 
ee Pn la Pr r 
U 0 
