OF AN INFINITE ANGLE. 263 



we for instance say that Cos co = 0, the selection of this particular value would be equivalent 

 to narrowing the hypothesis respecting x, as it would restrict a; to be an odd multiple of - , and con- 

 fine its variation to the terras of the series -, — , — , ; similar observations may be made 



2 2 2 •' 



respecting Sin oo . 



20. It is also very important to remark that Sin ax and Cos ax do not cease to be functions of 



a when x approaches eo . 



/I + Cos2.rYj . 

 For since Cos a- = ± I I , it appears that Cos.t; has an ambiguity of value of which 



Cos 2a,' does not partake. We may follow out tliis mode of reasoning to shew that Cos* and Cosoii- 

 have not the same number of corresponding values, and that if the value of one of these were given 

 the other would not be determinable from it except in an ambiguous form. Whatever indetermin- 

 ableness attaches itself then to Cos x when x approaches co , the same, and also another kind of, 

 indeterminableness belongs to Cosax at the limit. We are then particularly to take notice that 

 Cos OS derived from Cos x may not be written for Cos x derived from Cosax. Much error has 

 arisen from want of attention to this caution. Also Cos ax cannot be considered independent of a at 

 the limit /t? = co , inasmuch as it is subject to two causes of indeterniinateness which are distinct from 

 each other. 



21. Having thus given my reasons for considering that Cos co and Sin co have not definite 

 values, it may be proper to examine the proofs which have been brought forward by those who have 

 used definite values for Sin co and Cos eo . The following is the most direct proof I know of: 



.-. f" Sin X dx - {j" + J^" + ff^ + ... ad infinitum) Sin x dx ; 



.-. 1 - Cos 05 =2 — 2 + 2 — ... ad infinitum = 1 ; 



.-. Cos o: = 0. 



To this proof there are two objections, either of which is fatal to it. In the very first step it is 

 assumed that eo is an integer multiple of tt. For this assumption there is certainly no autho- 

 rity, neither is it compatible with the indeterminate nature of the symbol c© in the left-hand 

 member of the equation. The next error is made in the summation of the infinite periodic series 

 2 - 2 + 2 — ..., which I have shewn in the previous articles of this memoir cannot be equal to 1. 



22. As the reader may wish to have a further proof of the error of principle involved in 

 the first step of the above investigation, let him see the effect of a different distribution of co into 

 parts in the following process of reasoning, in which the question of summation of series is avoided. 



( r"- r-' r^' \ 



/^"Sinxdx =1 I ^ ■'■//■'■*' J i" ■*■ ad itifinitum] Sinxdx 



3 

 = -(1 + 0-1 + 1-1-0-1 + ) 



= - [[' + j„' + hw + ad injinitum] Sinxdx 



3 r" 

 = - I Sin m dx ; 



.•, _^* Sin^prf* = 1 - Cos eo = 0, .-. Cos os = i. 



