466 ■ Jjord 'Ra.jle\gh 071 the Character of the 



negative, and the negative sign when x is positive. The order 

 of integration is then to be changed, so as to take first the 

 integration with respect to x ; and finally a is to be supposed 

 to vanish. Thus 



27r'.{ct>{x)y = Um.r^ r rV""/iOO/i(«')]cos.^K-«) 



+ COS X [itf -\-u)\ du du' dx. 



Now 



so that 



I. 



e Q.o^iixdx=^ 



2a 



Thus 



Of the right-hand member of (19) the second integral 

 vanishes in the limit, since u and u' are both positive quan- 

 tities. But in the first integral the denominator vanishes 

 whenever ?/ is equal to u. If we put 



u' — u = az, du' = adz, 

 then, in the limit 



Jo a^+{u'-uy - J _ -i:^^ - '''■^^^^^ 



V^\<f>{x)Ydx=\^Jf,{u)Ydu. . . . (20) 

 If /sC'^) be finite, we have, in lieu of (20), 



yy<i.{x)Yda^= ^ Jj{/i(«)r+ {fMYyu. . (21) 



In M. Gouy's treatment of this question, the function 4>(x) 

 is supposed to be ultimately periodic. In this case /(«) 

 vanishes whenever ?/ differs from one or other of the terms of 

 an ai'ithmetical progression ; and the whole kinetic energy of 

 the motion is equal to the sum of those of its normal com- 

 ponents, as in all cases of vibration. The comparison of this 

 method with the one adopted above, in which all values 

 of n occur, throws light upon the nature of the harmonic 

 expansion. 



It is scarcely necessary to point out that vibrations started 



