476 
in which 
D 1 
w=\" . dx \ , dt Pry. 
« For the case y==a@, we must change @, in (B), to 
1 
w'=\* dv | df P.: 
o o 
and for the case y= 5, we must change it to 
; _ ; 
a =\"_% \ wat Pir 
«For values of y >5, or <a, the second member of the 
formula (B) vanishes. 
“If F,, although finite, were to receive any sudden change 
for some particular value of y between a and 3, so as to pass 
suddenly from the value F" to the value F', we should then 
have, for this value of y, ‘ 
\o dt \. d& Pir ty Fr =@ EF 4+0"F". 
By changing Pz to cos 2, we obtain from (B) the celebrated 
theorem of Fourier. Indeed, that great mathematician ap- 
pears to have possessed a clear conception of the principles 
of fluctuating functions, although he is not known to have 
deduced from them consequences so general as the above. 
« Again, another celebrated theorem is comprised in the 
following :— 
b ie 
Fy=o is ( a o Feb aes \ a & a de r2) ; (©) 
in which, the function @ is defined by the conditions 
ey 2Qnx-Kax 
ann | az P= y dz Pr; 
o 
2nxr— xr 
yis>a, <b; and no real root of the equation 
i dz P.= 0, 
except the root 0, is included between the negative number 
a—y and the positive number —y, nor are those numbers 
