438 On the Fundamental Modes of a Vibrating System. 

 whether the end be clamped, free, or supported. Hence 



(pN-^fuiafesO, .... (11) 

 Jo 



from which we infer that any two normal functions correspond- 

 ing to different periods are conjugate. 



This process may perhaps seem as simple as could be expected ; 

 but a little consideration will show that in the derivation of (11) 

 we have in fact retraced the steps by which the ordinary differ- 

 ential equation was itself proved, and that the true foundation of 

 (11) is the variational equation from which we originally started. 



In the equation referred to, namely 



8 V + §p'y'8yd% = 0, 



SV is a symmetrical function of y and 8y } as may be seen either 

 from its form in the present problem, or by the general theorem 

 proved below. Suppose that y refers to the motion correspond- 

 ing to a normal function u, so that y -\-p' 2 y = 0, while By is pro- 

 portional to another normal function v ; then 



oT 



=p* j puvdx. 



Again, if we suppose, as we are equally entitled to do, that y 

 varies as v and Sy as u, 



h V =p n I puvdx ; 



and thus, as before, 



(p h2 — p 2 )\ puvdw=0. 

 Jo 



lip andy are different, 



£V= f 'titufcsO (12) 



The symmetrical character of 6T is a simple consequence of 

 the fact that V is a homogeneous quadratic function of the co- 

 ordinates. If we suppose that 



V-*{ll}«+...+{12}?Mv*-'..., 



and let yfr 1 become ^ + Afa &c, we find 



AV={ll}^ 1 A^ 1 +... + {12}(^ 1 A^ 8 + ^ 2 A^ 1 )+... 

 + i{lH(A+ I )»-+ . .; + {12}Af ,Af 2 + . . . , 

 or, passing to the limit, 



SV={H}^h+ • • • + {12} W^ + ^atyi) +.. . , 

 which is Si symmetrical function of ^j &c. and 8yjr l &c. 



