352 Prof. Challis on the Motion of a small Sphere 



that I proposed to consider in this communication (which is, in 

 fact, the main object of it), viz. to ascertain the proper mode of ap- 

 plying the above-mentioned equations for determining the motion 

 of a small sphere free to obey the impulses of the undulations of 

 an elastic fluid. Here I shall have occasion to point out certain 

 modifications of the reasoning in the latter portion of Part III. 



The equations applicable to the present inquiry, so far as 

 regards & first approximation, are those which are called (8) and 

 (e) in Part II., viz., 



da- , dV 2U dW W . , . , , 



di + ik + -+^ + --*ote=o w 



These equations embrace all cases in which the motion is sym- 

 metrical with respect to an axis, U and W being respectively the 

 resolved parts of the velocity along and perpendicular to the 

 radius vector r drawn from a point of the axis and inclined to it 

 by the angle 6. Also to the first approximation the equations 

 are the same, whether the origin of r be fixed or moving. Let 

 us take the case of a disturbance of the fluid produced by a small 

 smooth sphere vibrating with its centre always on a given straight 

 line, and suppose the fluid to be incompressible and of unlimited 

 extent. Since the sphere impresses motion only in the directions 

 of the radii, if the motion be transmitted freely to an unlimited 

 distance, the lines of motion at each instant are prolongations of 

 the radii, and consequently W = 0. This is also very nearly 

 true for a compressible fluid of great elasticity if the radius of 

 the sphere be very small. This solution of the problem of the 

 simultaneous motions of a small sphere and the surrounding 

 fluid I have elsewhere supported by arguments which it is un- 

 necessary to cite here, my only reason for adverting to it being, 

 that in former investigations I passed from this solution to the 

 case of undulations impinging on a sphere at rest, by conceiving 

 motions equal and opposite to those of the sphere to be impressed 

 both on the sphere and on the fluid, and that I have recently 

 been led to question the correctness of this principle. If the 

 motion of the fluid relative to the sphere were actually the same 

 in the two cases, it would follow that the motion produced by 

 the reaction of the sphere at rest (which coexists with the rest of 

 the motion) is such that the surfaces of displacement are concen- 

 tric with the sphere, and the velocity at any point of a given 

 surface varies as cos 6. Now although, as is indicated by the 

 arguments above referred to, this may be true for a moving 

 sphere, the lines of motion always diverging from the moving 

 centre, and the motion of a given particle being consequently 



