Prof. Challis on the Principles of Hydrodynamics. 237 



tigation requires ; that it is justified by conducting to a value of 

 V, which clearly makes udx + vdy + lodz iutegrable ; that it is also 

 justified by conducting to a definite and unique law of action of 

 the parts of the fluid on each other, for such, it may be presumed 

 a priori, the law of that action must be. 



I propose now to exemplify the use of equation (10.) in a few 

 simple instances. 



Example I. Let the arbitraiy disturbance be a function of the 

 distance from a fixed centre : it is required to determine the 

 motion, the fluid being of indefinite extent. 



Let R be the distance at which the disturbance takes place, 

 and let the arbitraiy velocity Vj at that distance be a function of 

 the time; then since for this case ?•=?•', we have 



and 



V R2 



r^ 



This is the same result that would have been obtained by merely 

 using the equation (8.), and supposing the velocity to be a func- 

 tion of the distance from a centre. Although that course, ac- 

 cording to the argument here maintained, would be incorrect, it 

 happens in this instance to lead to no contradiction like that 

 which was pointed out (p. 36) in the analogous problem for a 

 compressible fluid. 



Example II. Suppose a smooth sphere of given radius to move 

 with its centre on a fixed straight line in a given manner in fluid 

 of indefinite extent ; required the motion of the fluid. 



The sphere being smooth, impresses velocities on the fluid only 

 in directions perpendicular to its surface. The velocity at any 

 point not on the surface is derived from the velocity at the sur- 

 face solely by the action of the parts of the fluid on each other. 

 Hence, according to the general residts already obtained, the 

 orthogonal trajectories of the surfaces of displacement are at each 

 instant straight lines, being in fact the prolongations of the radii 

 of the sphere, and the velocity on any one of these lines varies 

 inversely as the square of the distance from the centre of the 

 sphere. Also the surfaces of displacement are concentric sphe- 

 rical surfaces, of which the centre of the sphere is the common 

 centre. The motion is thus completely determined. It remains 

 to show tliat this motion satisfies the equations (8.) and (3.). 



Let K be the radius of the sphere, a the distance of the centre 

 from a £L\ed point in the line of the motion, and V, the velocity 

 of the centre. Then if that point be the origin of rectangular 

 coordinates, and the line of motion of the centre of the sphere 



