OF AN INFINITE ANGLE. 267 



31. It is not necessary to examine other instances of definite integrals the values of which, as 

 they have hitherto been obtained, I believe are not to be relied upon. They involve either the 

 notions that Sin oo = Sin (a . co ) = 0, Cos eo = Cos (a . eo ) = ; or else depend upon the sum of 



the series 1 - 1 + 1 — being = 1. The classes of definite integrals free from one or other 



of these errors are very few in number, not including some of those which analysts have evidently re- 

 garded with especial favor. It will be evident, if what has been written in the preceding pages 

 be allowed, that nothing could be more troublesome than the very general adoption of and 03 

 as limits of integration when trigonometrical quantities are involved. The expansion also of 

 functions in the form of series of multiple angles seems in very many instances to be attended 

 with much uncertainty, on account of the fact that Sinw.i' and Cos w.r become discontinuous 

 when w is 05 : and Fourier's celebrated theorem, that any function whatever can be developed in 

 a series of Sines and Cosines of multiple arcs, I regard as being fallacious in all cases where 

 the coefficients do not converge to zero as n becomes eo . As an instance, I have no doubt that 



A is not equal to 1 + Cos a; + Cos2.t7 + Cos 3a? + for any value of a; whatever. But this is 



too wide a field to enter upon in this paper, the object of which is to shew that Sin co and Cos eo 

 are not definite quantities, and that Sin (a co ), Cos(aeo) are functions of a. 



32. Perhaps it may be proper to add something in explanation of what is said in (26), 



1 . 1 r ■" Sin a a; ,,..,».» 

 respecting the integral / dx, that it is such a function oi a as is constant for ordinary 



■'o ^ 



values of a, and changes sign with a. This requires that a distinction should be allowed between 



arithmetical values and symbolical forms ; and such a distinction must be allowed, if any operation 



with respect to a is to be performed on the expression / dx. An example will best 



J^ a? 



explain what is meant. 



In Fourier's Theory of Heat, we find the equation 



— = Cosy - 1 Cos 3y + ^ CosSy — 



This equality is established (pp. l6^ — 174) by a method which is remarkable for its exhibiting 

 no symptoms of the existence of failing cases : and hence it is with surprise we read soon after, that 

 the left-hand member changes its value when y is comprised between certain limits. Guided 



by the investigation which Fourier gives of the sum of the series Cosy — \Cos3y+ we 



could have had no suspicion that the result is erroneous in any case ; yet it is manifestly erroneous 



when y lies between — and — . Hence the inference is plain, that the value — is not sym- 

 bolically correct, because it does not contain y, of which the proper form is obviously a function. 

 The author, at page 208, proves that 



J tan-' [ . °^5j =e-'Cosy-le-''Cos3j/ + ^e'^'CosSy - 



And consequently, admitting the propriety of putting x = 0, we obtain 



\ tan"' (2 00 Cosy) = Cosy — ^ CosSy + 1 Cos Hy - 



Now from this it is obvious that -^tan"' (2eo Cosy) is numerically = — for non-critical values of 



V, whenever Cos y is positive ; and equal to - — numerically, whenever Cos y is negative. 



M M 2 



