36 



when m is infinite : and it should create no surprise if conclu- 

 sions, deduced fi-om this assumption, prove to be absurd. 

 Bearing this in remembrance, let us take the series 



x 



- — sin re — ^ sin 2a; + ^ sin 3 a; — &c. 



first given by Euler, and which is known to be rigorously 

 true for all values of a; below tt.* 



From this series the following results have been deduced 

 by differentiation, and they have been pretty generally re- 

 ceived into analysis : 



\ =. cos a; — cos 2x + cos 3a; — cos 4x -)- &C' 

 = — sina; + 2sin 2x — 3sin 3a; -f- 4sin 4x — &c. 

 = — cosa; + 2^cos2a; — 3"^cos3a; + 4-cos4a; — &c. 

 and, generally, 



= cos X — 2^" cos 2a; + 3^" cos 3a; —4"" cos 4x + &c. 



= sina; - 22"+' sin2a; + 32«+isin3a;-42"+isin4a; + &c. 



so that putting a; :z in the first of these, and a; = - in the 

 second, we have 



= 1 - 22" 4- 3^" - 42« + &c. 



0=1^ 32«+l_^ 52n+l _ pn+i _,. ^^ 



results which are all inadmissible ; because, from the outset, 

 it is assumed that 



dsinmx . , dcosinx 



— ; — - := m cosmx, and — ; = — m sin mx ; 



ax ax 



though 7n be infinite. 



In reference to the preceding results, Abel justly asks : 

 " Peut-on imaginer rien de plus horrible que de debiter 

 _ 1 _ 22» + 32«_ 42« + &c. 



ou n est un nombre entier positif ?"t 



* It will be shewn, towards the close of this Paper, that it is true for all 

 values up to tt inclusive. 



■}■ CEuvres Completes, tome ii. p. 266. 



