NOTE ON THE MOTION OF WAVES IN CANALS. 279 



A more perfect agreement with theory than this could scarcely 

 be expected. Had the formula , /-fp = v been used, the errors 

 would have been much greater. 



The theory of the motion of waves in a deep sea, taking the 

 most simple case, in which the oscillations follow the law of the 

 cycloidal pendulum, and considering the depth as infinite, is 

 extremely easy, and may be thus exhibited. 



Take the plane (xz) perpendicular to the ridge of one of the 

 waves supposed to extend indefinitely in the direction of the 

 axis y, and let the velocities of the fluid particles be independent 

 of the co-ordinate y. Then if we conceive the axis z to be 

 directed vertically downwards, and the plane (xy) to coincide 

 with the surface of the sea in equilibrium, we have generally, 



p d<f> 



- dx* dz* 

 The condition due to the upper surface, found as before, is 



-r- -- ~ . 

 y dz d? 



From what precedes, it will be clear that we have now only 

 to satisfy the second of the general equations in conjunction with 

 the condition just given. This may be effected most conve- 

 niently by taking 



?f 9 



$ = He x sin - (v't x), 

 A< 



by which the general equation is immediately satisfied, and the 

 condition due to the surface gives 

 2-7T , 2 



where X is evidently the length of a wave. Hence, the velocity 

 of these waves varies as Vx, agreeably to what Newton asserts. 

 But the velocity assigned by the correct theory exceeds Newton's 

 value in the ratio VTT to \/2, or of 5 to 4 nearly. 



