(2) 



of Laplace's Differential Equation of the Tides. 389 



86 (M), Airy (87)], are 



• 9 /» d^i , n . n A dP a di/i — e) 



sm 2 ^ + 2n sm cos -± = -■ ^~ -^-y-,—. 

 dv dt r 2 dty 



where r denotes the earth's radius, n the angular velocity of its 

 rotation, g the force of gravity at its surface, and e the "equi- 

 librium tide-height " at time t, and colatitude and longitude 6 

 and <f> ; that is to say [Thomson and Tait's ' Natural Philosophy/ 

 § 805], the height at which the water would stand above the 

 mean level if it were so placed at rest relatively to the rotating 

 solid that it would remain at rest if the disturbing force were 

 kept constantly what it is in reality at time t. 



3. Laplace remarks that the general integration of these 

 equations presents great difficulties; and he confines himself to 

 a very extensive case, that in which 7 is a function of latitude 

 simply, and is the same in all longitudes. In this case the com- 

 plete integration is to be effected by assuming 



A=Hcos (at + sty),-^ 



% = a cos (at + sty), > (3) 



7] = b sin (at + sty) , J 



provided the disturbing force is such that 



e=Ecos (at + sty), (4) 



where H, a, b, E are functions of the latitude, of which E is given, 

 and H, a, b are to be found by integration of the equations. 

 With this assumption (1) bis and (2) give 



d(ya sin 0) , TT ^ . /M% 



a*a + 2na sin cos 0b = 9 - ^~^> 

 r dd 



g, . COStf 



a'b + 2na -. — -x a 

 sm 



_ g s(H-E) 

 r sin 2 



Putting, in these, 



H-E=w, 

 we find 



du 2ns cos 

 _ g d0 a sin 

 a -" T o- 2 - 4rc 2 cos 2 • 



2n cos du su 



-771 + 



A=- 



g a sin d0 sin 2 # 

 r o- 2 -4n 2 cos 2 J 



(6) 



en 



(8) 



