172 Mr. D. D. Heath on the Dynamical Theory of 



no doubt be proportional to j- (the rate of change in v), and so to 



dy d(0 



■~ (C) and to the horizontal force, and therefore will not affect 



the general form or position of the wave. The friction on the 

 bottom, on the other hand, varies directly with v, and so alters 

 the form of the effective force, and necessitates a reconsideration 

 of the problem. 



If a simple wave-form be compatible with the existence of this 

 friction, the ridge must necessarily lie to the eastward of the 

 moon ; for (3) the velocity at that point must be forward and 

 a maximum, and the effective force must therefore vanish. Now 



at the ridge the pressure-force ( —g ~t~ i =0; and the force of 



friction is eastward ; therefore the moon's force must there be 

 westward and equal to it. Similar reasoning shows that the 

 lowest point must be westward ; but it does not seem to be pos- 

 sible to adjust the general relations of forces and velocities every- 

 where to our conditions, except on the assumptions, 1st, that fric- 

 tion is proportional to the velocity, which is known to be ap- 

 proximately true in slow motions, and, 2ndly, that the effect is 

 to check equally all the water in each vertical column as if it were 

 all directly subject to the action of the sea-bottom — which can 

 hardly be true, though probably the resulting calculation ex- 

 hibits something closely resembling the case of nature ; for it is 

 the average forward velocity on which the form of the wave 

 depends. 



On this assumption, taking our origin at an indeterminate 

 distance B to the eastward of the moon, and putting, as before, 



y = cco$2co, we have (C) v= — ccos2«; and if/ be the coeffi- 



K 



cient of friction, its accelerating force at &> is — - — cos 2o>. 



On the other hand, the moon's force is now — 2gHsm2((o — 8), 

 or 



— 2yH icos 28 sin 2co — sin 28 cos 2o> J- . 



The second term of this force may be made everywhere to 



neutralize the force of friction by making 2yH sin 28—- — , which 



/c 



at once shows that c will now never become infinite. And then 



we have left an extraneous force, which differs from our former 



one only by being multiplied by cos 28. The conditions will 



therefore be fulfilled by making c==— — —. ^ — . Combining 



these two equations, squaring their terms, and observing that 



