46 
7 : ; 7 
5 = sing — 4sin 27 + 4sin37 — &e. — 0, 
resultat absurde.” 
Now the formula, agreeably to the general principle here - 
affirmed to be in fault, does really comprehend the limiting 
case x7, as well as all the cases up to this; for when « 
reaches this limit all the signs of the series become plus ; and 
as it is known that 
lta+st+2 +t &. =@, 
the series presents a particular case of 0 x» ; which it is wrong 
to declare to be 0, in contradiction of its legitimate interpre- 
tation, =) on the left. Thiserror has led Abel into other mis- 
takes of consequence: thus, at page 90 of his first volume, he 
says that the function 
“sing — dsin2¢+ 4sin3¢— &e. 
a la propriété remarquable pour les valeurs ¢ = 7 et p= —7 
’étre discontinue.” And at page 71 the same erroneous view 
has induced him to animadvert upon a certain principle of 
Cauchy, which the true interpretation of the matter would 
have tended to confirm, 
Fourier, Poisson, and many other modern analysts, have 
also made similar mistakes in their general investigations re- 
specting series. Thus, to quote Professor Peacock as to the 
views of the former, 
ee 4 6 
66 ay OSE == 191 a — sin4z : 2 ‘ 
cosz = “l5 sin 2a + zp sina + By sin6 a2 + &e. | 
a very singular result, which is, of course, true only between 
the limits 0 and 7, excluding those limits.”* 
The series is, however, true including the limits: for when 
a = 0, the signs are all plus ; and, as it is easily shewn that 
2 4 6 
Tea gi gp Be Oe 
* Proceedings of the Third Meeting of the British Association, p. 257. 
