Prof. Challis on a Mathematical Theory of Tides. 21 



preliminary considerations respecting the general hydrodyna- 

 mical equation 



*)-(x-©> + (T-(*)> + (. -(*))* 



in which the impressed accelerative forces in the directions of the 

 axes of coordinates are X, Y, Z. It will be assumed that 

 Xdx-j- Ydy + Zdz is an exact differential (d~F), so that 



e>+e)*+(?)*-w-»)- • ® 



This equation proves that the left-hand side of this equation is 

 an exact differential in all cases, if only the impressed forces be 

 suclt as to make Xafo? + Ydy + Zdz an exact differential. The 

 condition of iutegrability is therefore satisfied both when each 

 element moves as if it were solid, and when the motion in each 

 element is characteristically that of a iluid. We have seen that 

 in the latter case udx + vdy + wdz is always an exact differential, 

 and that in the former the same differential quantity is in- 

 tegrate by a factor. It thus appears that the left-hand side of 

 the equation (S) is made integrable by putting {d<f>) } or else 

 X(dyjr), for udx + vdy + wdz. 



In the cases in which (dp) may be substituted for udx + vdy + wdz, 

 the integral of the equation (8) is, as is well known, 



^F-^ + y+ftO, 



an arbitrary function of the time, which may also contain con- 

 stants, being added, because, as the differential is complete, the 

 integration may be taken from a fixed point of space in the fluid 

 to any other point in the same. 

 It may here be remarked, since 



-it© 



and since the integral 11-7-K 5 may be taken from anyone point 



to any other of the fluid, that the value of F undergoes no change 

 in passing from point to point of a given surface of displacement 

 at a given time^ because in that case the variation ds in the di- 

 rection of the motion vanishes. Hence we may infer that if 

 udx + vdy + wdz be an exact differential, the function F has the 

 same value at all points of the same surface of displacement. 



We may now proceed to apply these preliminary considerations 



