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



added suggestion, that, the moon not being stationary, we may 

 expect such a shape (lagging behind perhaps, and not quite fully 

 formed) to accompany the moon. And it is sometimes said that 

 with a perfectly open sea and without friction this shape would 

 be formed vertically under her. 



But it is easy to show that these anticipations rest on preca- 

 rious assumptions. For if the moon were suddenly removed, 

 the sea would not thereupon resume its spherical form, but the 

 waters would oscillate to and fro, and, may be, flow in some de- 

 terminate direction, until the energy belonging to its disturbed 

 state should be dissipated by friction. When, therefore, the 

 moon, instead of being entirely removed, shifts its apparent 

 place, we have to consider not only its instantaneous action 

 tending to form a spheroid under it, but also the effect of the 

 shape, velocities, and pressures due to its past action and still 

 subsisting. 



The true problem of the tides in its simplest form is to find, 

 if possible, some state of disturbance, some combination of shape 

 and velocities, such that, once supposed to exist, these combined 

 causes will exactly keep it up. 



2. The maximum of the moon's disturbing force, as above 

 described, may be the 10,000,000th part of gravity at the earth's 

 surface; and the problem may therefore be treated as one of 

 small perturbations, and the secondary effects neglected where 

 they come into competition with the primary ones. Still the 

 general discussion of the motion in our actual seas has proved 

 unmanageable; and the only case which has received a com- 

 plete and satisfactory treatment is that of a sea supposed con- 

 fined to a narrow canal in which the motion is conceived as only 

 vertical and longitudinal, and in which the depth is uniform. 

 I propose to take this case first, and to begin with a canal run- 

 ning round the equator, supposing the moon vertical over it and 

 moving uniformly in her orbit, so that her apparent motion will 

 be also uniform and somewhat less than that of the earth's 

 rotation. 



It is shown, in Herschel's l Astronomy ' and elsewhere, that 

 the moon's differential force goes through all its phases of mag- 

 nitude and direction twice over as our eye travels over this cir- 

 cuit, repeating with close approximation in the opposite hemi- 

 sphere the same changes which it exhibits on the side where 

 itself is. We propose, then, to inquire whether a wave-shape re- 

 peating itself in the like spaces can, consistently with the laws 

 of fluid-motion and the forces in action, be so adjusted, both as 

 to the elevation and depression and as to the oscillatory motions 

 of the particles, as to be persistently propagated at the rate at 

 which the moon apparently moves. 



