SECT. IV.] SOLUTION OF EQUATION OF VIBRATION. 411 



F(OL) being any function, whatever the limits of the integrations 

 may be. This value of v is merely a sum of particular values. 



Supposing now that t = 0, the value of v must necessarily 

 be that which we have denoted by/(#, 0) or &amp;lt;f&amp;gt; (x). We have 

 therefore 



(f) (x) = IdoL F (a) \dq cos (qx qx). 



The function F (a) must be determined so that, when the two \ 

 integrations have been effected, the result shall be the arbitrary I j 

 function &amp;lt;j&amp;gt; (x). Now the theorem expressed by equation (.6) shews J 

 that when the limits of both integrals are oo and + GO , we A 

 have 



Hence the value of u is given by the following equation : 



I /+ [+*&amp;gt; 



u = ^ dy. &amp;lt;/&amp;gt; (a) I dq cos ft cos (qx qa). 



Air J -so J -oo 



If this value of u were integrated with respect to t, the &amp;lt; in 

 it being changed to ^Jr, it is evident that the integral (denoted 

 by W) would again satisfy the proposed differential equation (d), 

 and we should have 



W= 27rj d *^ W fa - 2 sin & cos (&amp;lt;l x - 2*)- 



This value W becomes nothing when = 0; and if we take the 

 expression 



dw 



dw i r + 



&quot;^ = 2^rJ 



we see that on making t in it, it becomes equal to -^ (x). 

 The same is not the case with the expression -j- ; it becomes - 

 nothing when t = 0, and u becomes equal to &amp;lt; (x) when t = 0. 



It follows from this that the integral of equation (d) is 



1 r +x r +ao 



# = I da&amp;lt;t&amp;gt;(a) \ dq cos ^^ cos (qx qz) + W= u + TF, 



^7T J -oo J - x 



and 



1 . 



j Sin Q&quot;t COS (QX 



i r&quot; 1 &quot; 00 r +ao 1 



Tr= g- I rfaA|r (a) I dq ^ 



