132 Mr. Boor on the Analysis of Discontinuous Functions. 
as g 1S positive or negative. Now if, under the integral sign in the first member, 
we make £’ and & to vanish, we have 
2 Nx 
\ da cos (qx cos = eee 
eee) 
a 
¥ da cosqa m==9) 
f = , 5) 
a” F ne ( 
5 2T (7) cos > 
the upper or lower sign being taken, according as ¢ is positive or negative. The 
first member of this equation is a limiting integral of the second class, and is to 
be regarded as merely an abbreviation of the first member of (4). Taken in any 
other sense, the equation is certainly not true. 
If in (1) we make g = 1, we have 
i dx cos(a) a" = T(2) cos (x 5) 
en) —* (" dxcos(a)a" ; 
cos(n 5) wo 
and if we assume this as the universal definition of P(7), and regard the integral 
in the second member as a limiting one of the first or second class, according as 
7 1s positive or negative, we may deduce the law of continuity of T(n). For 
compare the first members of (1) and (5) with the general form ‘ da cos(x)a*™, 
in which / may be positive or negative ; we have, in the former case, G=Nn=& 
and the equation (1) becomes 
( dx cos(.r) a‘ = (7) cos (5) ; 
0 
in the latter case, g = 1, n = 1 —J, and the equation (5) becomes 
a” me pacar Me Ne ike EI 
\ dx cos(w) a! = 2r(1—Icos((1—1) 5) = 2r(1—l)sin(/5)° 
Whence 
Tv 
T(/)cos 3) = 2r(1—l)sin( 0%) 
