183 



with reference to the interpretation of the second member, and 

 to the changes of value which it undergoes in passing the 

 limits, as the preceding one. 



By an integral transformation, Mr. Boole then deduces 

 the third, or Fourier's theorem, involving two signs of inte- 

 gration, viz. : 



] noD poo _f^^ 



/(x)-=:-\ \ dadve cos (av— xv)f(a). (3) 



7rJ_Qo Jo 



He remarks, that when this theorem is written in the 

 form, 



J\x) = - \ \ dadv cos {av — xv)J{a), 



we must attach to the symbol \ a meaning different from its 



Jo 



r»co 

 ordinary one, and understand by \ dv<lt{v), the limit of 



Jo 



i 



00 _J^y 



dvi. <p{v), for decreasing positive value oik. Mr. Boole 



proposes to designate an integral of this kind as taken in a 

 limiting sense, and he observes that some anomalous results 

 have been obtained by writers who have neglected the distinc- 

 tion here implied. Thus, from the equations 



V dx cos X — 0, y rfa; sin .r = 1 , 



Jo Jo 



r»<X) 

 in which the sign \ has been used in its limiting sense, have 

 Jo 



been deduced, by taking that symbol in its ordinary sense, 

 and integrating without reference to the factor understood, 



£~ , the incompatible conclusions, 



cos CO n 0, sin 00 = 0. 



Mr. Boole remarks that, by a converse error, other writers 

 have been led to infer the incorrectness of Fourier's theorem. 



VOL. III. Q 



