20 Prof. Challis on a Mathematical Theory of Tides. 



equations are identical, inasmuch as both apply to motion which 

 is peculiar to a fluid. As this expression for V has been obtained 

 prior to the consideration of any particular case of motion, it 

 must be applicable in all cases, and at every point of the fluid 

 mass at all times. Consequently it partakes of the character of 

 a general integral, but gives the value of the velocity, in any par- 

 ticular case of the disturbance of the fluid, only through an infi- 

 nitesimal portion of fluid, for which the radii of curvature of the 

 surfaces of displacement maybe supposed to have, at a given in- 

 stant, the same positions. If such a portion be bounded laterally 

 by planes coincident in direction with lines of motion, any trans- 

 verse section of the small included element will be proportional 

 to the product r'r". Hence, if the velocity and transverse section 

 be V] and m\ at one end of the element, and V 2 and m\ at the 

 other, we shall have Y l m\=.'Y 2 m\. Then passing to the next 

 element, and supposing r' and r" to change in magnitude and 

 position either continuously or discontinuously, but not per sal- 

 tum, we have similarly ^^n\=Y 3 m\', and so on. This reasoning 

 proves that if the whole of the fluid be divided into filaments 

 of indefinitely small transverse section, having their lateral boun- 

 daries everywhere coincident in direction with lines of motion, 

 and the value of V^ at a certain transverse section of a fila- 

 ment be given, the value Vm 2 at any other transverse section is 

 also given. This inference is evidently consistent with the prin- 

 ciple of the constancy of mass, and is independent of the form 

 and position of the filament, which have to be deduced from the 

 given conditions of the particular case of motion. 



The general integral of the equation (j3) may, as Poisson has 

 shown, be expressed under a form which involves arbitrary func- 

 tions of impossible quantities. This analytical result might give 

 the means of arriving at that form of the general integral which 

 is obtained above by a different process ; but actually it renders 

 the integral unfit for application to the particular circumstances 

 of a case of motion. In fact, as the equation (j3) takes no account 

 of the impressed forces, its general integral cannot possibly ad- 

 mit of specific applications to given cases of disturbance of the 

 fluid, and can only have the general signification which I have 

 just indicated. This remark will be further illustrated by the 

 proposed method of solving the problem of the tides. 



In the subsequent analytical reasoning a differential is put in 

 brackets to show that the differentiation has relation to coordi- 

 nates only ; a differential coefficient in brackets is a complete 

 differential coefficient, with respect to time, of a function of co- 

 ordinates and the time; all other differential coefficients are 

 partial. This being understood, I proceed next to adduce some 



