4f0 Profr^J. |i,. Y^^u^gpf^ihe Evahiatio7i.(^ 



for the quantities themselves, and all their changes, are essen- 

 tially independent of one another. Each value of cosfi, as A 

 passes through its changes, gives rise to a distinct series of 

 cases; and we are not to assume the law of continuity in pass- 

 ing from one of these distinct series to another, through the 

 variation of cos 9: if we wish to consider the continuous re- 

 sults of this latter variation, A must be regarded as fixed. 



I make these observations in case it might be imagined that 

 there was anything anomalous in the circumstance that [1.] 

 gives for the limiting value of the series in every case of 

 cos 9, except the single case of cqS/(3|=^1j^ ^yhen the value is 



. u,, 2(i--A)^~.2(i.-^A)T:,;^,^?^"*:".^~.4fi^Hf 



It thus, I think, appears that Mr. De Morgan is not ifif' 

 error, as Mr. Moon supposes, in affirming [1.] to be thfe'* 

 limit of the proposed series when A = I. The real error, sd^ 

 frequently committed in analysis, consists in confounding"' '^' 



— H- cos 9 + cos 2 3 + cos 3 9 + &c. ad infinitum - ,<i 



with the limit of 



— + A cos 6 + A?cos 2 9 + A^ cos 3 9 4- &c. ad infinitum^ 



and calling [1.], when A = 1, the sum of the former; an^i 

 this belongs to the same class of errors as those discussed vck^ 

 the preceding part of this communication. 1 reserve a nior^) 

 detailed examination of the influence of them for a future ocj-^ 

 casion. • jj 



Poisson, in the 12th volume of the Journal de VEcole Poly^jfi 

 teclmique, a source which I have not as yet been able to consultif j 

 has, I believe, entered upon an examination of the series whichfj 

 form the subject of the present paper ; and I think he connects { 

 the extreme case of each with the continuous series of cases 

 as I have here done. But, from all that I can collect respect- 

 ing his memoir, he falls into the errors here noticed, and 

 which it has been my principal purpose to point out and correct, .1 

 by showing the essential distinction between an isolated series 

 and a series connected to a chain of others by the bond of|(» 



continuity. The symbol — prevents the condition implied in'*' 



this connexion from being obliterated in the extreme case ; 

 and I think that this symbol, together with the symbol V — 1*..* 

 employed for an analogous purpose in my former paper, in , 

 the August Number of this Journal, might, with propriety, be.' 

 called symbols of continuity. By disregarding the influence^ 



