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



where I is arbitrary and 



n*c 

 2 = 



Solving this. 



c— 



2qlB 



2qlg — n* 



the sign of which varies with the sign of q, and with the mag- 

 nitude of qL If q be positive, or the depth diminish with 6, 



n 2 

 there will be low water under the moon, unless ql> n~. Ifg 



: .... . ty 



be negative, c is positive, or there is high water Under the moon 



whatever the depth. To have the diurnal and semidiurnal tides 



both of the form we have been considering, we must have q = l. 



If <7=0, c=0; that is, with a sea of uniform depth, and 

 the moon stationary in her orbit, we should have no visible di- 

 urnal tide ; only small oscillatory motions corresponding to the 

 values of u and v. 



In the case we have been considering, where rn = n, the angular 



COS 



velocity v=VjCosa) is of the form, a .—. — ^coso>, which becomes 



infinite at the pole. But the linear velocity v sin 6 — a cos 6 cos w ; 

 and if we combine this with u=a sin co, we get the whole linear 

 velocity =«\/l— -sin 2 0cos 2 o>; and the tangent of the inclina- 

 tion of its direction to that of the moon's meridian = = . 



sin g> 



Hence at the pole the actual velocity is #, directed westward 

 along the meridian perpendicular to the moon's meridian 

 (Airy, 103). 



I am not aware that the extremely special character given 

 to these theorems of Laplace, and particularly as regards the 

 diurnal wave, by making m = n, has been before remarked upon. 



11. The friction, which makes the wave take an oblique posi- 

 tion with reference to the moon, necessarily implies a reaction on 

 the solid earth ; and it is now admitted on all hands that one 

 effect is a retardation of the earth's rotation. But the mode and 

 the measure of this action seems not yet quite settled upon. 



Mayer, in a paper translated in this Journal (1863), argues 

 (on false grounds I think) that there must be, and states that in 

 fact there is a perceptible westward current in spite of causes 

 which tend to eastward motion ; and he assigns to this prepon- 

 derant motion the effect of diminishing the rotation. 



But then he goes on to assign another cause, viz. the form of 

 the wave, as presented to the attracting force of the moon; and 

 proceeds to calculate the amount "in the same manner as that 

 employed in computing precession/' — a process which is based on 



