( 561 ) 



If now we put ?i = in the equations (T) and (//) we find 

 2 rrc 

 - I F^ G^dioz= ^ a, a\ 4" «i «'i + «* « « + 







/F. 6r, doj = aj a\ + a, a', + • • • 



thus 

 1 







which is the theorem in question. 



1 C 1 



4. From the preceding formulae we may also deduce the values 

 of several interesting series. For, if the series for /{x) and ^ (.i') are 

 given, and the integrals 



f{co) (fi (to) COS 710) dio and | ƒ (co) (p (w) sin na> duj 







are to be found, the values of the series in the second members of 

 the given equations may be determined. To show this, we shall make 

 the following application of the formulae (1), (2) and (3). 

 Suppose ƒ (.^•) = X, then 



n 4 /^cos X cos Sx cos bx 



+ 



and 



2 jt\V 3" ' 5^ 



^sin X sin 2x sin Sx 



2 T" cos nn 

 Vl„ = — I tu* cos no) do) = 4 







J- 



Jt J n' 







2 T'^ ^ 2n cos nn 4(1 — cos njx) 

 (è„ = — I to" sm Jio) day = — — 



nj n nn 







Now the formula (1) gives, because 



2I„ = è a„' + a,^ + a,' + a,' + . . . 

 31, = i a^' + a^aj + a^a^ -f aja^ + . . . 

 21, = a^a^ + a^a^ + a^a^ + a^a, -f- . . . 



3i, == a^a, + "2" + «iS + «B«» + «6«ii 4- • • . 



