846 
Fu) =0 and F(a) F, (5) =o 
hold good. 
These lemmas having been demonstrated, the proof of formula (1) 
for any arbitrary value of mn will be easy, viz.: we shall demon- 
strate the proposition for 7=7,-+7,, supposing that it has been 
proved for n=n,, for n=n, and for n=n, -+n,—1, where n, 
and n, represent two arbitrary positive integers; as the proposition 
in $ 2 has been proved for n =1, the validity for n = 2, 3, 4... 
etc. respectively, follows from this argument. 
Let Flu) be the function with parameters mm and n, + 1, for which 
relation (1) has to be proved; we introduce (and according to the 
preceding lemma this is possible) the function /’(w) with parameters 
m and n,, the function F',(w) with parameters m and n, and the 
function (u) with parameters m and n, + n,—1, so that we have 
Fo) n,/n,! SF (OF WF e(t 
u) = —— — 2F (dE, — |+ C u 
(rn, +-7,)/ du hd a 
and consequently 
x nin! x 
2 Fe) =A MOED O SF 
u—2 (0, 4-7): dd’ <« u— 
u= dd' =! u— 
As relation (1) holds good for n= n, + n‚—1, consequently for 
the function |F,(w), we have 
en DO 
(oll 
r : frat eae 
BF le ay pee 
co | log © 
u =| 
and as according to our proposition, (1) holds good also for n =n, 
and for n= n,, i.e. for the functions F, and F,, we have, accord- - 
ing to the second lemma of this paragraph 
1 1 
bm(n, +n Jem am For! oo flv) 2m ge mna? 
=) CHA 3 2 a dE) is ; 
dd'<« hn,!n,llog a ee 1 log # 
dd'=l pam) U" 
so that we conclude 
1 4 1 | 
zs bmm a ma+n—1 a0 i v ee amd? 
=, Ku) = = aS I) Wp : 
psy h(n, |, lege = as log x 
= = pit 
u=l vaml 
therefore formula (1) has been proved for all positive integers 7. 
