293 
the upper or lower sign being taken, according as q is posi- 
tive or negative. Assuming as the definition of I'(n), the 
equation 
x 
T(n) = vi dx cos (x) a"—", 
whether 2 is positive or negative, and regarding the integral 
in the second member as a limiting integral of the first or 
second class, according as x is positive or negative, the author 
shews that, universally, 
T(n) T—n) = 
sinnw’ 
a theorem which is known to be true of IT in its ordinary de- 
finition when n lies between 0 and 1, but not otherwise. This 
‘ theory is further applied to explain the discontinuity of form 
which is apparent in integrals, the subjects of which become 
infinite within the limits of integration, with some other con- 
nected points. 
The paper concludes with an application of Fourier’s 
theorem to the solution of equations. It is proved that the 
value v of the definite integral 
. l a a 
a —\- ) da dv e"-2)"V—| F(a, vy —1) 
as 2r J-ax J—x ; 
symbolically expressed by the equation 
ee d 
. Q d\@ 
ae N cee: A as (-=) J (# &), 
provided that @=0, from which the following theorem is 
: deduced : 
— __—siIf. f(u) = & and ¢(2) be any function of « which makes 
——- F[p (@)] real, then 
eS d ; 
t « x d a Py 
F(u) on (- —) (Flo @—ae F’[o(x)]¢' (2), 
i _ which may be expanded in the form 
