262 Mr. EARNSHAW, ON THE VALUES OF THE SINE AND COSINE 



cannot be eqtial to zero. Hence the terras of this series at an infinite distance from the beginning 

 are subject to discontinuity, and cannot be made to approach 1 as their limit ; because if d differ 

 ever so little from zero there will always be a term so distant from the beginning as that n9 is 

 finite; that term and all following ones will not approach 1 as their limit. Consequently 1 — 1 + 1 - ... 

 is an isolated form of 1 — Cos9 + Cos 29 — ... 



15. It is not necessary to repeat, in reference to this series, what has been already said upon 

 1 — .V + a!" — ... ; it is sufficient to remark that all results are nugatory which have been obtained 

 upon the supposition of 1 - Cos + Cos 2 - ... approaching 1 - 1 + 1 - ... as its limit as 

 9 changes continuously towards 0. I might here add remarks in reference to the series 

 a^ + a,Cos0 + a, Cos 20 + ... + a„ Cos i'9 + ... parallel to the remarks in (12) and (13). 



16. Since it often happens that by integration as remarked in (13) a real equivalency is 

 established, it is not unusual to find such series cited as confirmations and verifications of the 

 propriety of the equivalency assumed to exist before integration. From what has been proved 

 above however it is evident that such verifications are of no value, and do not in any degree justify 

 the inference sought to be drawn from them. 



17- I come now to examine the limiting values (if such there be) of Sin x and Cos a; when 

 ■T approaches co . As a preliminary step it is proper to remark, that oo is an indefinite symbol : and 

 when it is said that a; approaches eo as its limit, we are not to understand that ,r approaches towards 

 some definite value, but merely that it approaches to a value of which we have no other property 

 than this, that it is greater than any finite quantity. Yet there is such a thing as a restricted oo . 

 Thus, if a? be an odd multiple of tt by the nature of its definition, this restriction will not hinder its 

 becoming infinite ; yet then the symbol 05 will be specific ; and accordingly it is possible that under 

 such a condition definite results in certain cases may be obtainable. 



18. The above remarks respecting the essential indefiniteness of the symbol cs will enable us 

 at once to reply to some questions which have been found perplexing. The question has been 



asked, is the series P^ - P.^ + P, - P^ + ad infinitum equivalent to the series (P, + A) 



— (P., + A) + (P-s + B) — {Pf + B) + ... ad infinitum? This has been rightly answered in the 

 negative ; but on erroneous grounds. The true reason is this : the terms A, B, C ... are introduced 

 in such a manner as necessarily involves the notion that co is an even number, and therefore it 

 creates an error unless it have been stipulated that 05 is an even number. As from the nature of 

 an infinite series no stipulation of this kind can be allowed, we are justified in saying that the two 

 series are not equivalent. 



19. If J? be defined to be a term of the series 0, 2, 4, 6 ..., then Cos wir = Cos 0" when x = (» ; 

 but if x be a term of the series 1, 3, 5, 7 ..., then Cos aiir = Cos ir wheu .r = 05 ; but if x be defined 

 to be a term of the series 0, 1, 2, 3, 4 ..., then it cannot be affirmed that x is an odd number, nor yet 

 that it is an even number. To say only that x is a whole number, is to express oneself in a way that 

 requires the result to leave the question as to whether x is odd or even undecided. Hence in this 

 case we cannot say that Cos eo = CosO", nor yet that it = Costt; but we must express the result in 

 such terms as leave undecided which of these two is the value of Cos co ; for to select one of them 

 and reject the other would narrow the restriction laid upon x by its definition, by deciding that it is 

 not only an integer, but that it is a specific integer. Hence then in this case Cosco = CosO" or Costf 

 indeterminably. 



This mode of reasoning can be extended without difficulty to the case where x is a continuous 

 variable, and it leads us to this result, that on this hypothesis respecting the nature of x, Cos eo 

 (derived from Cos x by supposing x to approach towards co ) is equal to the Cosine of any angle 

 from 0" to 2 7r indeterminably. When I say indeterminably, I mean to say that we cannot fix on 

 one of these angles and reject the others without violating the generality of the hypothesis : should 



