46 



- = sin TT — i sin 2 TT + i sin 3 TT — &c. =. 0, 



resultat absurde." 



Now the formula, agreeably to the general principle here 

 affirmed to be in fault, does really comprehend the limiting 

 case .r=r7r, as well as all the cases up to this; for when w 

 reaches this limit all the signs of the series become joto; and 

 as it is known that 



1 + i + i + i + &C. = 00, 



the series presents a particular case of x oo ; which it is wrong 

 to declare to be 0, in contradiction of its legitimate interpre- 

 tation, -, on the left. This error has led Abel into other mis- 

 takes of consequence : thus, at page 90 of his first volume, he 

 says that the function 



" sin — ^ sin 2 ^ + ^ sin 3 ^ — &c. 



a la propriete remarquable pour les valeurs ^ = 7ret^= — tt 

 d'etre 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, 



4 r 2 4 6 ~| 



" cosA- = - |_-p^ sin 2« 4- — sin 4 A' + ^ sm6w + &c.J 



a very singular result, which is, of course, true only between 

 the limits and tt, excluding those limits."* 



The series is, however, true including the limits : for when 

 ,T z=L 0, the signs are all plus ; and, as it is easily shewn that 



-o + ^ + A + ^''- = "' 



* Proceedings of the Third Meeting of the British Association, p. 257. 



