494* 



Mr. Moon on Fourier's Theorem, 



principle that " v/hat holds generally holds in the limit," is 

 not true except in the case of continuously increasing or de- 

 creasing functions. Let X represent the equivalent for the 

 indefinite integral 



f' 



s 9y yP cos 7^ y 

 in formulae (B.), then the definite integral 



s~^yyP cos r y 



f 



consists of two parts, l^y^^ and Xy=o5 ^^e first of which 

 varies continuously with regard to g', and the latter does not. 

 Now of the latter it is perfectly true that the equation 

 _g-gy.p.(jo— 1)...3.2.1 



X, 



=0 — 



cos(ry + p+l .5), 



which holds for all finite values of g', holds also when 5- = 0. 

 But it is not true that because when q is finite and positive 

 Xy=oo vanishes, it therefore vanishes when 5^ = 0. The prin- 

 ciples of limits are here inapplicable. For the limit of a quan- 

 tity or ratio is defined to be that quantity or ratio to which it 

 continually approximates, but which, although its difference 

 from it may be made less than any assignable quantity, it 

 never actually reaches ; and it is obvious that none of these 

 characteristics obtain in the case of the function X^y^g^, con- 

 sidered as a function of q. 

 In like manner the equation 



*J 



■qx 



sm rx 



= tan 



-1 



which holds generally when q is finite and positive, does not 

 hold when §» is = 0. 



Having thus, as I trust, established the position that the 

 sine and cosine of an infinite arc are quantities which may 

 assume any value between + 1 and — 1, it remains to say a 

 few words on the geometrical interpretation of those symbols. 



