488 Mr. Moon on the Symbols sin oo ajid cos oo, 



function are wholly erroneous, and that Fourier's theorem, 

 by which the function itself was suggested, is untrue. 



1. As to the function. The function which Mr. De Mor- 

 gan finds is the following : — 



— A d(tif cos(ti{v—x),vdv. 



It is easy to show that this 

 _ _1_ /**T J's'moo{l—x) __ {l—x)s'mco{l—a;)—xs'm(tix\ 

 TTi/O L «> w J 



_ x /**, fsmco{l—x) sinu)x'\ 



~ 'TTc/O \ W C*J J' 



Mr. De Morgan assumes — / dw = -— , which is 



not the case, as will be seen when we come to consider the 

 way in which that result is deduced. 

 We have 



^ n 



cosr^6""g'^= g . c, ; .... (u.) 



q^+r^ ^ ' 



whence, integrating with respect to /*, we have 



f 



xJ 



svnrx _„^ _i r 



6 



-?^ = tan-i— ,. . . . (/3.) 



a conclusion perfectly true so long as q is finite, but not other- 

 wise, for when q = the equation (a.) does not hold ; and in- 

 stead of it we have 



X 



sm 00 



cos rare""* = , 



r 



•ox 



where (for anything yet proved to the contrary) sin oo may be 

 any quantity between + 1 and — 1 ; 



'sinr.r . , , ^ 



= sm CO log, r + C, 





^ 



from which we have 



C = log, . sin CO = CO . sin c» , 

 = any quantity whatsoever, 

 which demonstrates the impossibility of assigning a definite 



/^* sin Ic ui 



value to the function / dw, and hence Mr. De Mor- 



*y CO 



gan's interpretation, which proceeds upon the contrary hypo- 

 thesis, entirely fails*. 



* It might be argued, that as the equation (/3.) holds generally for all 

 finite and positive values of q, it must hold in the limit when q = 0, but 

 this is not true, as will be explained hereafter. 



