( 559 ) 



We shall now investigate the relations which exist between the 

 integrals A,,, Bn and the coeiTicients an, b,,, a'„, b'n. 



Substituting in An for (p (to) the series of cosines, we obtain 



2 T" 

 An = — I /(to) COS noi [4 a'g -\- a\ cos oy -\- a\ cos 2 (o -\- » . .] dio 







00 xj' 2 Z'" 



1 2 jtj 



=z ^ a\ ttn -\- 2 -— . — I ƒ (w) [cos {m -{- 7i) o) -\~ cos (m — n) to] do) 

 1 2 Jtj 







= è «'o «n + -^ -T" {<^m-{-ti + «m— ?i)- 

 1 -^ 



This equation may be written in another form ; for, because 

 d — p ^^r ci p , 



7)1 tï jyj 



-^ — — «/« — n ^ -^ ~^ öS/i — m ~r 2 -^ ^ wi <^/n — n 

 1 ^ 1 '^ n-fl 



or, putting m -j- ?i instead of vi in the summation from n -}- 1 to co 



00 ^' n 00 



•V" ^" 1 'ST' ' 1^ 1 V ' 



-^ -— ttm—ii — 2 -*^ ^ »i ^« — '« 12-^ '^/n '^ m-|-7! • 



I '^ 1 1 



Hence 



An^= ^ S a\nan—m -\- h ^ («'m «;;i+n + «m «'m+n) • • • (/) 

 1 



If now we substitute in An for t/) (to) the series of sines, we have 

 An^=^ — I ƒ (to) C9S 7? lo \h\ sin CO -\- h\ sin 2to + • • ] ^^ 







"6',„ 2 r . 



= ^ -T- . — j f{oj>) Ysin [m -\- n) o) -\- sin {m — n) cd] dio 



1 ^ ^J 







= -2" — - (6m-|-„ + ^,n— n) 

 I ^ 



or, as Z)'— ;, = — hp 



n oc 



An := — i 2 h'm hn-m + i ■2' (6',„ 5^_|_„ + 6,„ 6',„-|_„) . . . [II) 



In the same way we find 



2 r^ 



^n = — I /(to) sm nto [4 a'j -[- «'i cos to -f «', cos 2to + . . ] c?to 







°° a' 



= è *'o ^n + -2" -^ (èrn-^-n — &;„—„) 



or, after a slight reduction 



