Mr. Booue on the Analysis of Discontinuous Functions. 135 
v= (— FE II) aadoe va (a,e) 
by a known form of Fourier’s theorem. 
Since, in this investigation, the function f(a,e¢) may be discontinuous, and 
may be supposed to vanish for all values of a which are not included between 
given limits, it is obvious that the value of the more general integral, 
Syl 
Spe 
(\ dadv eV = f(a, €*), (6) 
will be expressed by the formula 
axe 
v=(—+)*/(2,2), 
when zx lies within those limits, supposed to be real, to which the integration 
relative to a extends, and that it will be O when z lies entirely without those 
limits. 
2. The above premised, let us consider the expression 
Lye a= = 
v= g\ dadvesrv5 w(a), (7) 
which may be written in the form 
ae = dadve? 2) xy elfel-# a) "YAR (q), 
Qn JJ_2 
@ '(a) denoting an arbitrary function of a, to which the inverse form is here 
given for the convenience of a subsequent transformation. 
Let ¢'(a) =a’, thena=¢(a’), da = ¢'(a’) da’, therefore 
1 : = Ais es 
Vv = aZN) daldnce var eee V5 w [p(a')] g(a’) 5 
the value of which, supposing the limits of a’ to be real, and x to lie within those 
limits, is 
v= (— 3)" ee" e[oalo (8) 
the function $() being arbitrary. 
