Equations of Hydrodynamics. 441 



It is, however, to be observed that, as was before stated, ^ may 

 be supposed to include a term which is a function of the time. 

 For instance, if the motion be along straight lines drawn from a 

 centre, and be a function of the distance from the centre, we 

 shall have with respect to a given particle, ^r=r— /(/) = C, r 

 being the distance of the particle from the centre at the time t, 

 and the value of fit) depending on the given circumstances of its 

 motion. 



9. Assuming that the equations (1), (2), and (3) are necessary 

 and sufficient for the determination of the motion of a perfect fluid, 

 before applying them to that purpose three considerations of a 

 general character, which it will be important to bear in mind, will 

 now be stated. (1) The indications of the analysis are coexten- 

 sive with the circumstances of the motion ; so that there is no 

 circumstance of the motion which has not its analytical expres- 

 sion, and no analytical circumstance, or result that is not per se 

 impossible, which does not admit of interpretation by circum- 

 stances of the motion. (2) Any analytical result obtained with- 

 out taking into account all the three equations, must admit of 

 interpretation relative to the motion, although the application of 

 such interpretation will be subject to limitations. (3) Analytical 

 results which admit of interpretation relative to the motion prior 

 to the consideration of particular disturbances, indicate circum- 

 stances of the motion which depend only on the quality of the 

 fluid, and on necessary relations of its motion to time and space — 

 such, for instance, as is the circumstance of the uniform propa- 

 gation of motion in an elastic aeriform fluid. These three 

 remarks will receive illustration as we proceed. 



10. In the Philosophical Magazine for March 1851, I have 

 obtained by two methods the following equation as the solution 

 of Frop. VIIL, viz. 



This equation is deduced exclusively from the principle of con- 

 stancy of mass combined with that which is the foundation of 

 the third general equation, namely, that the directions of motion 

 are normals to continuous surfaces. But that equation is not 

 used in deducing it ; and as it involves no consideration of pres- 

 sure or accelerative force, it is wholly independent of the first 

 general equation. In the Number of the Philosophical Maga- 

 zine for November 1853, the equation (4) is employed in obtain- 

 ing expressions for V and p under the following conditions. The 

 motion is supposed to be central, and to be a function of the 

 distance from the centre ; and any three spherical surfaces of 

 radii r — 8r, r, and r + Br being drawn, the quantity of fluid 



