159 



And the equation of continuity becomes: 



— m2Xi sin m2^+ni2FiC0s m2^— S2-CY2 sin S2^+S2F2 cos S2^+ • • '" 



+ D(dFi/d,r)cos m2/+D(dZi/da;)sin m2t+D{dV2/()x)cos Sii 



+D(dZai/c)x)sin S2^ • • • =0 



or: 



(DdVildx-\-m2Yi) cos uiit-}- {DdZi/dx— 1112X1) sin mot 



+ (DdV2/dx+S2Y.2)cos S2t+{DdZ2/dx-S2X2)sin S2t-\~ ■ • • =0 (195) 



Equations (194) and (195) are satisfied by all values of t, if: 



bXi/dx-\-Yn2Zi/g=0 bX2/c)x-^ 82^2/0=0, etc. 



bYi/c)x-m2VJg=0 bY2/dx-S2V2/g=0, etc. 



DdV,/dx+m2Yi = Z)dTydx+S2 1^2=0, etc. 



DbZJdx-m2X,=0 DbZi/bx- 82X2=0, etc. 



320. Expressions jor the components oj the tide. — An examination of 

 these equations shows that the variable coefficients for each of the 

 tidal and current components are related by the equations: 



bXlbx+aZjg=0 (196) 



bYlc)x-aVlg=0 (197) 



Z>dX7dx+aF=0 (198) 



D(iZlcix-aX=0 (199) 



in which a is the speed of the component. 

 From equations (196) and (197) 



Z= - (g/a) dX/dx V= (g/a) dY/dx (200) 



whence: 



bZ/bx=-(gla)d'X/()x' bVldx={g/a)()'Y/dx' (201) 



Substituting these expressions in equations (198) and (199): 



b'Yldx'+ (a'lgD) Y=0 (202) 



b'X/dx'+(a'/gD)X=0 (203) 



Evidently the solution of equation (203) will afford also the solution 

 of equation (202). 



Placing for convenience, gD^c', and multiplying the terms in ec|ua- 

 tion (203) by 2dX/bx this ecjuation becomes: 



2(dX/d.r) (d-T7dx-) +2 (aVc^) A'd.Y/dx=0 (204) 



The integration of which gives the equation: 



(bXlbxy-+ (a'lr)X'=K' (205) 



in which K^ is a constant of integration. 



