ae 
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. : 
JK(2) = -(" \" da dve cae (av— av) f(a). (3) 
He remarks, that when this theorem is written in the 
form, 
1°02 Ca 
fiz) = a) ) da dv cos (av — av) f(a), 
TI_~Yo 
@ 
we must attach to the symbol a meaning different from its 
0 
io) 
ordinary one, and understand by ) dv $(v), the limit of 
0 
a : ee 
) dve™  (v), for decreasing positive value of k. Mr. Boole 
0 
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 
ie 8) @ 
) dx cos x = 0, ) dx sinx = 1, 
0 0 
@ 
in which the sign J has been used in its limiting sense, have 
0 
been deduced, by taking that symbol in its ordinary sense, 
and integrating without reference to the factor understood, 
kx Y ; , 
¢  , the incompatible conclusions, 
cos o = 0, sino = 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 
