36 
when m is infinite: and it should create no surprise if conclu- 
sions, deduced from this assumption, prove to be absurd. 
Bearing this in remembrance, let us take the series 
5 ena seit ae + psind 2 — Re. 
first given by Euler, and which is known to be rigorously 
true for all values of x below z.* 
From this series the following results have been deduced 
by differentiation, and they have been pretty generally re- 
ceived into analysis : 
1 = cosa — cos 2x + cos3ax — cos4a + &e. 
0 = —sina + 2sin 2a — 3sin3a 4 4sin4a — &e. 
0 = — cosa + 2?cos2a— 3?cos 3a +4 4°cos4a — &e. 
and, generally, 
0 = cosa — 2% cos22 + 3" cos 3a —4" cos 4a 4+ &e. 
0 = sina — 2"+'sin2a 4+ 3%+!sin3a—4"t'sinda 4 &c. 
so that putting « = 0 in the first of these, and « = 3 in the 
second, we have 
0 = 1— 243" — 4" 4 &e. 
O = 1 — B™ +14 541 _ 741 4 &e. 
results which are all inadmissible; because, from the outset, 
it is assumed that 
dsin mx 
dx 
though m be infinite. 
In reference to the preceding results, Abel justly asks : 
«¢ Peut-on imaginer rien de plus horrible que de débiter 
O=1— 2" 4+ 3°%— 4% 4 &e. 
dcosm2z é ‘ 
= —msinmez; 
dx 
= mcosma, and 
ou n est un nombre entier positif?”’t 
ee 
* It will be shewn, towards the close of this Paper, that it is true for all 
values up to x inclusive. 
+ GEuvres Completes, tome ii. p. 266. 
