Deep-sea Tides, and the Effect of Tidal Friction. 169 



4. We have now to inquire whether the forces which are 

 called into play in a system of disturbance, having the charac- 

 teristics above described, can keep it permanently in action. 

 And first in the case of a free wave, as Mr. Airy denominates it 

 when there is no extraneous force like the moon's. 



(a) The earth rotating with a velocity (n), a force n 2 is, in the 

 undisturbed state, employed in maintaining the circular motion, 

 and the force producing weight is less than the whole attractive 

 force of the earth by this quantity, which, however, is very small. In 

 the disturbed state the centripetal force required becomes (n— v) 2 , 

 or alters by — 2nv nearly; and its direction is normal to the 

 now slightly inclined surface. Resolving the force thus set free 

 into the vertical and horizontal directions, the latter component, 

 having the sine of a very small angle as a factor, is of the second 

 order of small magnitudes, and may be neglected. 



(b) The waved surface of the sea exerts an horizontal attrac- 

 tion on each of its own particles. But we may omit the consi- 

 deration of it, because it obviously depends on the mass of fluid 

 in the surface, and becomes less, indefinitely, as we suppose our 

 canal narrower and narrower, while the other accelerating forces 

 remain unaffected. 



(c) We are thus left with only vertical forces, and these sen- 

 sibly equal to gravity (g) acting on each particle ; and the only 

 effective force producing horizontal motion is the difference of 

 pressures before and behind each molecule. And as we neglect 

 the vertical effective forces as of the second order, the vertical 

 pressures are the same as if the fluid were at rest; i. e. the pres- 

 sure at any height (h) above the bottom is the weight of the 

 column above it, or g(ic-\-y—h). Therefore the accelerating 



force forwards ( — j~) is — 9 -j-, and is proportional to the for- 

 ward and downward slope of the surface. And this is indepen- 

 dent of the depth at which the particle is ; so that if the veloci- 

 ties of the particles in the same vertical column be equal at start- 

 ing, they will remain so, as we just now assumed. 



dv 

 We have used -j, to represent the change in time of the ve- 

 locity of the particles at a given place. The forward accelera- 

 ting force on a given particle, whose velocity when at « is v, is 

 the rate of change from co and t to to-f vdt and t-\-dt) i. e. it 



dv dv 

 is -T- -f -=- v. But the second term is the product of two small 

 at aco r 



quantities, and may be neglected ; so that we have 



dv dy ■■ 



