NEWTON S PRINCIPIA. 391 



X of the particle which was situated at the distance x from 

 the origin to be represented by 



that then all the conditions required for a possible motion 

 will be satisfied. If this expression agree with the 

 initial conditions and all the other circumstances of the 

 motions, it will represent the actual motion. The whole 

 of the reasoning is, step for step, the same as that for the 

 motion of sound in a tube. 



Since this form of X is independent of the depth, the 

 water will be agitated to the bottom, and the particles once 

 in a vertical plane will always remain in one. If then two 

 planes be taken at a distance dx when the fluid is at rest, 

 the mass of water between them will always remain the 

 same. But this mass before the motion was h dx, and at 

 the time t it is 



the last term is very small, for both y and X are small, and 

 therefore may be rejected ; hence 



= a m h . cos (nt mx). 



If we consider any element dx of the fluid, the forces 

 that make it move are the pressures on its two ends. These 

 are, by hydrostatics, 



and g(y + dy + z), 



where z is the depth of the element beneath the mean 

 surface. Hence the moving force is 



c c 4 



