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



But we have seen that for our wave to remain permanent in form 

 with a rate of propagation «, we must have 



dv _ « 2 dy 



dt fc dco 



Therefore we must have 



a 2 =g/c. (E) 



And if we start a wave by giving to the surface any shape what- 

 ever consistent with our -assumptions, and then give to the par- 

 ticles of water in every column a westward velocity a / -y t the 



pressure-forces thus brought into play will be such that, after 

 the time dt, the shape and the velocities will be the same, reck- 

 oned from a point x/g/c dt westward of the origin, that they were 

 at first, reckoned from the origin itself; or the wave will be pro- 

 pagated without change at the rate \Zgx, — a rate altogether in- 

 dependent of its contour, but increasing with an increase either 

 of the depth of the canal or of the effective force of gravity. 



An increase of tc produces its effect through the condition 

 of continuity, causing a greater rise or fall for the same hori- 

 zontal motion of the particles. The effect of an increase in the 

 effective force of gravity would be by increasing the pressure- force 



( — g —- ) for the same slope of surface. 



If, then, this, last effect be produced by the direct application 

 of an extraneous horizontal force, leaving the vertical force as it 

 was, or even increasing it or diminishing it by a small quantity 

 of the first order, the result on' our formulas will obviously be the 

 same, as the only operation of the vertical force is to produce the; 

 horizontal one. 



5. We see then the solution of our problem. 



The moon's differential horizontal force is found to be (when 



on the equator, and reckoning co from her meridian) — 7Tn3 snl 2°V 



M and D being her mass and distance. Imagine, then, a wave 



dy 

 once started in which — shall be everywhere proportional to 



sin 2o>, i. e. let its surface be defined by the equation y= ccos2o>, 

 where c may be positive or negative. As a free wave, it would 

 slip away from under the moon eastward or westward according 

 as \/gic is less or greater than the apparent rate of the moon's 

 motion. But as v. forced wave, the moon^s force is at the first 

 instant everywhere combining with it, either by addition or sub- 

 traction, according as c is negative or positive, increasing or di- 



