Mr. Lubbock on Bernoulli's Theory of the Tides. 461 



Where the blank is left, the London results are too irre- 

 gular to be entitled to any dependence. The London correc- 

 tion in time is much greater than that for Liverpool. 



If x 9 y,z are the coordinates at the end of the time t, of 

 any element dMof the ocean; p the density of the fluid; Xd Jk/, 

 Yd M 9 ZdM the components parallel to the coordinate axes 

 of the force acting upon dM; and if the components of the 

 velocity of the element d M, in two positions, which it oc- 

 cupies successively, are u, v 9 w, and u -f u' d t 9 v+v'dt, 

 W+tt/ d t ; the differential equation to the surface of the ocean 

 will be 



(X-u) dx + (Y-i/) dy + (Z— w/)dz = 0. 



(See M. Poisson's Traite de Mecanique, vol. ii. p. 669.) 



If the forces arise from attractions or repulsions directed 

 towards fixed or moveable points, 



Xdx + Ydy + Zdz = dV. 



This condition obtains in the forces which produce the 

 tides. 



. du du du du 



v = -7— -f u -j h v —j h •■ 



dt dx dy dz ' 



Generally, if Xdx + Ydy + Zdz is the exact differential 

 of any function V with reference to the variables x 9 y, z, and 

 if v! d x + tidy + w'dz may be neglected, the surface of the 

 fluid is given by the equation 



V = constant. 



That is, the surface of the fluid assumes the same form at 

 any given instant, as it would do if the forces then acting 

 upon each particle were invariable in magnitude and direc- 

 tion. It seems worthy inquiry in what cases this approxima- 

 tion is admissible. 



If r be the distance of the sun's centre from that of the 

 earth, £ the sun's zenith distance, m the mass of the sun ; if 



■ M 

 the same quantities accented refer to the moon ; and if -^ 



is the force of gravity, R being the distance of the fluid ele- 

 ment d M from the earth's centre ; then in the problem of the 

 tides, 



M \ Rcost, 1 \ 



W -R~ m \ r* {i2 2 -2ri2cos?+r 9 }*J 



JjRcosX 1 -t 



