SECT. IV.] TWO DEFINITE INTEGRALS. 413 



have proposed them to the attention of geometers. They shew 

 how oscillations are propagated and set up through the whole 

 extent of the lamina, and how the effect of the initial displace 

 ment, which is arbitrary and fortuitous, alters more and more as 

 it recedes from the origin, soon becoming insensible, and leaving 

 only the existence of the action of forces proper to the system, the 

 forces namely of elasticity. 



407. The results expressed by equations (1) and (2) depend 

 upon the definite integrals 



I dx cos ce 2 , an d I dx sin x* ; 



f-f-oo r-f-oo 



g = I dx cos cc 2 , and h = I dx sin a; 2 ; 



J ao J - ao 



let 



and regard g and h as known numbers. It is evident that in the 

 two preceding equations we may put y + b instead of x, denoting 

 by b any constant whatever, and the limits of the integral will be 

 the same. Thus we have 



g = P*dy cos (y* + Zby + b 2 ), h = ( ^ dy sin (y 2 + 2by + 6 2 ), 



J 00 J 00 



= f di I cos ^ cos 2 ^ cos ^ ~~ cos ^ 2 s * 



J I sin y 2 sin 2by cos 6 8 - sin y 2 cos 2by sin b 2 ) 



Now it is easy to see that all the integrals which contain the 

 factor sin 2by are nothing, if the limits are &amp;lt;x&amp;gt; and + o&amp;gt; ; for 

 sin 2by changes sign at the same time as y. We have therefore 



g = cos 6 a I dy cos y z cos 2by - sin b* I dy sin y* cos 2by ......... (a). 



The equation in h also gives 



h = id i S ^ n y * cos 2 ^ cos ^ + cos y* cos ^y sm 



J \ + cos y 2 sin 2by cos b 2 sin y 2 sin 26y sin 

 and, omitting also the terms which contain sin 2by, we have 



h - cos & 2 J dy sin y 2 cos 2by + sin Z&amp;gt; 2 / dy cos 2/ 2 cos 2by ........ (6). 



