380 Frederick Guthrie on Stationary Liquid Waves* 



waves, as was pointed out so many years ago by the brothers 

 "Weber, do not complete their orbital motion, but themselves 

 swing, like the bobs of elementary pendula, to and fro in short- 

 arc'd rhythmic sequence. Whenever the momentum and inertia 

 vary together (as in all cases of solid magnification — i. e. altera- 

 tion of size, conservation of shape, and either conservation or 

 alteration of density), the pendular law must be preserved, and 

 this whether the mass to be moved is wholly active, as in the 

 pendulum itself, or more or less passive, as the logan oscillation 

 of a common balance. 



In rectangular troughs it is easily shown that, if the surface 

 is plane, the pendulum (elasticity) or torsion law must hold 

 good, and that the force of restoration is proportional to the 

 linear measure of disturbance. If AB (fig. 5) be the level of 



A 



Fig. 5- 





^ — "~~~~^ 



c 



^^*^*~~~*~* 



JutSw 







H 



E 



the water in a rectangular trough and it is set in mononodal 



undulation, so that in one phase the surface is at C E D, the 



line B D, B F, or A C is half the amplitude. Consider all the 



water below C F to be without influence. The rectangle of 



water ACBF has become the triangle CFD. Join CB and 



E F ; their intersection K is the centre of gravity of C D F, and 



is J of C B from C, &c. Draw G H horizontal and K H vertical 



till they intersect. The centre of gravity of the rectangle 



ACBF has been raised by the change of form from G to K — 



that is, from H to K. The work in the displaced water is 



DCFxKH. 



BF 

 KH = KG jjpj, 



and KG=(f-i)BC=|BC, 



or 



KHr^yDF. 



