TEEMORS OVER THE SURFACE OF AN ELASTIC SOLID. 



25 



dotted curve in fig. 5. If a; be constant, the effect of increasing t will be to cause 

 this graph to travel uniformly from left to right ; and if we imagine that in each of 



Fig. 5. 



its positions the integral of the product of the ordinates of the two curves is taken, 

 we get a mental picture of the variation of v {) as a function of t, on a certain scale. 



For the greater part of the range of t, the integral will be approximately 

 proportional to the ordinates of the curve V(0), viz., we shall have 



(98), 



TT/Jibx ' \X/ 



in analogy with (95). But for a short range of t, in the neighbourhood or ex, the 

 statement must be modified, the dotted curve being then in the neighbourhood of the 

 vertical asymptote of the function V(#). Since the principal value of the integral is 

 to be taken, it is evident that as t approaches the critical epoch and passes it, v will 

 sink to a relatively low minimum, and then passing through zero will attain a 

 correspondingly high maximum, after which it will decrease asymptotically to zero, 

 the later stages coming again under the formula (98). 



Although the above argument gives perhaps the best view of the whole course of 

 the disturbance, we are not dependent upon it for a knowledge of what takes place 



VOL. CCIII. A. E 



