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



It is for those mathematicians to reconcile these conflicting results, who maintain that providing 

 the last limit of x be os it is no matter whether it be a specific oo or a general os. The dis- 

 tinction is of first-rate importance in periodic functions. I think I am fairly entitled to affirm that 

 specific values for Sin 03 and Cos co are obtained by such processes as that in (21), only because 

 those very processes assume at the outset a specific form of co . 



23. The next proof which I shall examine depends upon the principle of continuity "that 

 what is true up to the limit is true at the limit." It is as follows : 



Since fe'°' Sin xdx = (Cos x ■\- a Sin x) ; 



•' 1 + a' ' 



■'■ /^" e"°' Sin a;d.r = ,. 



■' \ + a' 



This being true for all positive values of a, no matter how small, is taken to be true in the 

 limit when a = 0, which gives (since e'"^' then = 1 for all values of x) 



j^ e'" Sin.rd.r = /," Sin xdx = 1 ; 



.•. Cos CO = 0. 



24. To this investigation I have two objections to bring forward. The step which assumes 



that e'"' = 1 for all values of x is not true at the limit x = oc , for however small a become ax 



will be finite and arbitrary or infinite when x = «> . Hence as we diminish a towards zero e"" 



approaches, not to 1 as its limit when x — <si , but to e'" an arbitrary value depending upon 



the relative laws with which x approaches co , and a zero. Now it is absolutely necessary in the 



above proof that for all values of x between zero and 03 , e""' should be equal to 1 ; and as 



this is not a true hypothesis, the proof fails. 



1 e'"' 



Again, it is essential to the above investigation that should be the value of - 



1 + a' 1 + a:' 



(Cos a? + a Sin a?) between the limits .r = 0, ,r = co . But this will not be the case unless e"'" 

 vanish when .r = co . Now I have just shewn that when a is made to approach zero e'"' 

 become e'" at the limit a! = co . This step therefore of the investigation is erroneous, and the 

 proof fails. 



Let us look at the first written equation in (23), and endeavour to answer these questions; 

 can e'" in the left-hand member be always = 1, and yet in the right-hand member = 0, when 

 X = CO ? If a; = CO make e""' = in the right-hand side, what can prevent the same being true in 

 the left-hand side, seeing that the values of x are simultaneous in both members ^ Here is a 

 plain contradiction of hypothesis in the two members of the fundamental equation the consequences 

 of which no explanation can remove : and as both hypotheses are required to be true together 

 to enable us to obtain the final result Coseo = 0, I conclude that this result is not proved to 

 be true. I think upon examination of the steps of the proof in (23) the reader will admit, 

 that it is conducted upon the supposition that, as x varies from zero to 03 , e~"' remains constant 

 on the left-hand, and decreases from l to on the right-hand. 



25. These are the usual proofs that Cos co = ; and it is not necessary for me to examine 

 more, as all that I have met with involve erroneous reasoning of a character similar to that 

 noticed in the two above given. Before concluding I wish however to notice one or two other 

 cases in which great caution is necessary in managing the symbols Sin eo and Cos eo. 



26. The first which I shall notice is / d,v, which has been said to exhibit some 



■J„ X 



