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



in which case we shall come to a differential equation of the first 

 instead of the second order. 



Be this as it may, the only results hitherto obtained of such 

 definite form as to illustrate general physical laws will appear, 

 and, I think, more clearly, by using the equations to answer a 

 question which the course of our previous investigations natu- 

 rally suggests, viz, whether a wave exactly representing the form 

 of the force be in any circumstances of sea-bottom possible ; and 

 if so, whether the law as to the inverse position of the wave in 

 shallow and deep seas will not apply. 



Let us then assume y 1 = csin 2 ^. 



The left-hand terms of the first two equations will then take 

 the form (C—gc) sin 6 cos 6, and — (C —gc) sin 6 ; and the left- 

 hand term of the last equation will be — 2mcsin 2 0. 



And now, without solving the equations, we can answer the 

 last part of the question. For if we obtain u x and v x from the 

 first two equations, they will each be functions of 6 multiplied 

 by (C —gc) ; and when substituted in the third equation (in its 

 expanded form), it will take the form 



d/c -p mc ~ 



wnere P and Q are functions of independent of either C or c. 



Now it is evident that /c, if not constant, must have symme- 

 trical values on each side of the equator, because the forces are 



IT 



the same in both hemispheres. Therefore, when 6— ^> 



dd u ' 



If, then, on making 0= -, the known functions P and Q do not 

 vanish, putting / for the depth at the equator, we have 

 . . . 7_ mc {Q\ 



and we can determine c in terms of / without knowing the law 

 of depth, only assuming it to be the right one. 



For a free wave of the same form this equation holds, making 

 C = 0. If, then, L be that equatorial depth for which the free 

 wave of this form would move on with the rate m, 





9 

 which is independent of the height of the wave. Substituting, 



