432 PROF. A. ('. IUXOX ON STl'KM l.loi: VIl.hK HARMONIC KXl'ANSIONS. 



rnfitcii of il,, li u itrc i-nnjiKjah- vmagin&riet, n<l <tr<- suc/t its to y/'re tin- 

 to 



1 J 



j f(x) i (jc) a- 2 -i (x} ... , (x)\ 3 dx, 



Jo p 

 tttdl IS. to 



.'o p [ i \ ( i 



Tin- raliif <>f this intiyral. <dl therefore also that of 



f 1 1 f "' 



Jo p L" i 

 r 1 1 



tend to zero as in is increased, if - (fxf dx exists. 



Jo p 



26. The integral 



f <a(t,x) dX 

 Jot) ~ U" X-X' 



tends to zero when B -> cc if x t, and thus it follows that 





To this expression the methods of Dr. J. MERCER ('Roy. Soc. Proc.,' A, vol. 84, 

 p. 573, and 'Phil. Trans.,' A, vol. 211, pp. 134.^) may be applied, hut his idea of the 

 bilateral limit cannot be used without some modification, since we have no reason 

 to believe that even at a point of discontinuity where f(x 0) exist their mean is 

 represented by the present expansions. 



