46] PARTIAL DIFFERENTIAL EQUATION. . 171 



the limits for x also, consequently we must have the factor of the 

 arbitrary 6y vanish, that is, 



108) 



tting 

 motion of the continuous string, 



o 



Putting = a 2 we have the partial differential equation for the 



which may also be obtained from the ordinary differential equa- 

 tions 84) by passage to the limit in an obvious manner. 



The passage from n ordinary differential equations to a single 

 partial differential equation when n is infinite is worth noting as a 

 type of a phenomenon of frequent occurrence. At the same time the 

 notion of normal vibrations gives rise to that of normal functions. 

 To find a normal vibration let us find a particular solution of 109), 



110) y = X(x) <p(t), 



where the two functions contain only the variables indicated. This 

 satisfies the definition of a normal vibration, since the ratios of dis- 

 placements of the different points are the same throughout the 

 motion. Inserting in the differential equation we obtain 



1-H\ -\rd*<P 9 d*X 



m > x -dw = a fd^- 



Dividing by Xcp we have 



1 d*y _ a* d*X 

 ~^ dt* ~X~di*" 



Since one side depends only on x and the other only on t, which 

 are independent variables, this can hold only if either member is 

 constant, say v 2 a 2 , where v is arbitrary. Thus we have the two 

 equations 



* = 



112) 



The first of these shows, like 77), that (p is a normal coordinate. 

 Its integral is 



113) (p = C cos (vat a), 

 the integral of the second is 



114) X = A cos vx + Bsinvx. 

 The normal vibration is accordingly represented by 



115) y = (Acosvx + B sin vx) COB (vat a), 

 the arbitrary constant C being merged in A and B. 



