440 PROFESSOR YOUNG, ON THE PRINCIPLE OF CONTINUITY, ETC 



whenever the second member is a converging series. Abel says, " Pour avoir les sommes de ces series 

 lorsque a = + 1 ou - 1, il faut seulement faire a converger vers cette limite * :" and he then writes 



i log (2 + 2 cos (f) = cos (|) - - cos 2 + ^ cos 3 (/) - &c. 



t^^ 1 I 11 



- log (2 - 2 COS (/>) = - cos - - COS 2 - - cos 3 (/) - &c. 



which he says, "a lieu pour toute valeur de <p excepte pour (/) = (2/-. + 1)t dans la premiere expression, et pour 

 (^ = 2M7rdanslaseconde." Now the second members of these equations are not the limits of the proposed 

 general form, any more than 1 - 1 + 1 - &c. is the limit of 1 - 1 + j:" - &c. A limit always implies continuity, 

 and is never exempt from the control of that principle : putting therefore the condition of continuity in 

 evidence, the preceding expressions should be written 



1 log (2 + 2 cos 0) = (l - ^) cos - i(l - ^y COS20 + J (' - ^J <:os 30 - &c. 



ilog(2-2cos0) = -(l-^)cos0-i(l-^ycos20-i(l-±)cos30-&c. 



which are true, whatever be the value of 0, for the series are always convergent. As tends to the values 

 excepted to by Abel, the series tend to infinity; which they actually attain when these excepted values 

 are reached, as the first members sufficiently show. We thus see that 



is infinite as well as 



, 1 1 1 « 



2 3 4 



\ -\- 71 



In like manner, from the development of log — - — , we should infer that 



-K'-^)*K'-^)"*K'-^)'-=' 



is infinite as well as 



and thence that 



is infinite as well as 



. 1 1 1 » 



.3 3.5V eo/ 5.7\ 05^ 



1 . 3 



12 3. 



H + +&C. 



1.3 3.5 5.7 



so that any of these diverging infinite series may be replaced by the corresponding dependent converging series, 

 and vice versa, without numerical error. And a priori considerations, in reference to this class of diverging 

 series, would lead us to the same conclusion. The equations [/if] are thus universally true without any 

 exception whatever. 



• (Euvres Computes. Tome i . p. 89. 

 Belfast, September, 1846. 



