130 Mr. Boots on the Analysis of Discontinuous Functions. 
conclude, therefore, that the second symbol of integration in Fourier’s theorem 
is not to be taken in its ordinary acceptation. 
The peculiar advantage of Fourier’s theorem over (5) and (9) is, that x 
enters into the second member only exponentially, through the cosine. 
5. Resuming the equation 
: UNC . 
WAC) = 2 i dad cos(av—xv) f(a), 
and writing the cosine in its exponential form, we have 
. ES (Ce v= WAP 
flay = HVS dado fer + eV f(a), (4) 
Now 
ey = 
t  7(n) 
n—} ete =I es dw ern — Vgt ( 15) 
Tiny 
the symbol \, being used in its extraordinary or limiting sense. Multiply the 
successive terms of (14) by those of (15) respectively, and we have 
S(&) _ 
F =o3- = 
= nT nr 
a \ dadudw { 0? ta W—1t_ e020 wt) =i} wy"? F(a), 
0 9 
and converting the exponentials into cosines and sines, 
Ae) = =S VA. \ dado cos(av— ro—tw+ — ia ey) (a). (16) 
Of course this theorem only enables us to represent discontinuity relative to «. 
Its advantage, with reference to ¢, is, that it introduces that variable under the 
symbol cosine. ‘The application is reserved for a separate paper. 
SUPPLEMENTARY INVESTIGATIONS. 
1. We have designated as extraordinary or limiting integrals those which are 
to be considered as the limits to which others approximate, as some quantity 
approaches to 0. Of this kind is the integral A dxcos(qx)2"', n being positive, 
