Prof. Challis on a Mathematical Theory of Tides. 23 



(dQ) for the left-hand side of the equation (8); and because 

 tidal motion is evidently such as is proper to a fluid, we have 

 besides (d(f>) — udx-\-vdy-\-wdz. 



It is readily seen that integration gives 



+ ^y (a? 8 (3 cosV-1) +2/ 2 (3 sinV-1) + 3xy sin2 fit- z*). 



Hence, since by integrating ($) we have p = ¥ — Q, + ^r{t) } after 

 substituting the values of x, y, z, the result will be found to be 



+ ^- 3 (3cos 2 \cos 2 ((9-^)-l)-Q + ^W. j 



As the proposed method of solution is necessarily approximative, 

 small quantities of the second and higher orders with respect to 

 the disturbing force, as also the terms containing k and &) 2 , will at 

 first be omitted, and it will afterwards be shown how the omitted 



terms might be taken into account. Hence, putting-j for Q, 



p=-Gr+~(3c OS ?\co^(0-vt)-l)- d -£+W). (,) 



It is next required to obtain from this equation the complete 

 differential of p with respect to t — that is, the variation of/? for a 

 given particle in a given indefinitely small interval of time. In 

 the differentiation for this purpose, r, X, and 6 will be considered 

 constant in the term containing m, because their variations 

 would introduce terms of the second order relative to the dis- 

 turbing force. Consequently 



Now let the resolved parts of the velocity in the direction 

 of the radius vector produced, and in the two directions at right 

 angles to this, westward and northward, be respectively w', v ! , 

 and w 1 . Then 



, dr dd> . . d6 dS , rdX dd> 



at dr at r cos \du dt rdX 



