of a Liquid Substratum beneath the Earth'' s Crust. 217 



and dv 



W =~dx Z (2) 



But we know (A) that ^-=-A 

 v J dt dx 



Wherefore from (1), 



dv _ot dy < 



dx ~~ k dx' 



dw _ a. d~y 



dx k dx 2 ' 



and also a. 



v =«y (3) 



By the same reasoning as that by which we have concluded 

 that — J = — a— we likewise conclude (B) that 



dw _ dio 

 dt ~ da? 



« 2 dy 2 . 



(4) The equations of fluid motion in two dimensions will 

 be, p being the density and p the pressure at Z: — 



(1 dp _ tfr tf« dv 



\pcte =X -dT- V dx~- W ck> ••••(!) 



) 1 dp „ dw dio dw , x 



f ~- -f- = L~ -rr— to- io— (2\ 



V dz dt dx dz y ' 



In our problem p is constant. Considering a tide formed 

 over a rigid bottom, the horizontal forces are the difference 

 between the moon's horizontal attractions at the point x, z, 

 and m a parallel direction at the earth's centre, and the' 

 friction. 



The moon's differential horizontal attraction is - ISSJsin 2co 



2D ' 



M being her mass, a the earth's radius, D the moon's mean 

 distance, and co the longitude west of the moon's meridian. 

 It will be negative, because it acts in an easterly direction. 

 Call it — jx sin 2co. 



_ The horizontal friction, if taken as proportional to the velo- 

 city v of the liquid (which is true of low velocities), / being 



the coefficient of friction, is fv, or (by (3) of § 3), /- y. When 



