282 NEWTON S PRINCIPIA. 



On this subject Newton has not many propositions. He 

 has shown an analogy between the motion of waves and 

 the oscillation of water in a pipe. 

 Let there be a column of fluid 

 of length I in the U pipe M A m : 

 let the water when at rest stand 

 at the level M m. Then if it be 

 raised above this level in one leg 

 to the height P, it will be depressed an equal 

 amount in the other, say to Q. The weight of water 

 tending to pull it back again will be equal to twice the 

 weight of the column P M ; that is, the force varies as the 

 displacement, and tends to pull the body back to its ori 

 ginal position. This case of motion has been investigated 

 in the First Book, and the result is, that the motion is 

 vibratory, the time of oscillation will be always the same, 

 and equal to 2 TT divided by the square root of the ratio of 

 the accelerating force at any distance to that distance. 

 But the moving force is twice the column P M, and the 

 mass moved is the column Z. Hence the time will be 



/2l 



**/-, 



or it varies as the square root of the length of the column 

 of water. 



Now, in waves the motion is carried on by the succes 

 sive ascents and descents of the water ; hence the time 

 that must elapse before any particle now at the top of a 

 wave will again be the top is 



where / must here be assimilated to half the length of a 

 wave. We cannot say they are equal to it, but they will 

 increase or decrease together. Newton assumes 



