to Mr. Airy's Remarks. 131 



For in the case of motion under consideration, if v^ be the 

 velocity impressed on the fluid by the vibrating sphere at the 

 extremity of the radius drawn in the direction of its motion, 

 the velocity impressed at the same instant at any point of the 

 surface, the radius to which makes an angle 6 with that direc- 

 tion, is in fact v^ cos 6, the sphere being supposed perfectly 

 smooth ; also the ratio of the two velocities, which is cos 6, 

 does not vary with the time. It is, however, undoubtedly 

 true, as Mr. Airy contends, that if the motion be of this kind, 

 the lateral pressures, namely, those which take place at the 

 surface of the sphere in directions perpendicular to its radii, 

 should be taken into account; and the explanation which Mr. 

 Airy desires will, I suppose, be given, if it be shown, as I am 

 about to do, that these pressures are taken into account in 

 my solution of the problem under discussion. 

 • It will be admitted that if the same accelerative force be im- 

 pressed at every instant on the sphere and on all points of the 

 fluid surrounding it, in the same direction, no motion of the 

 fluid along the surface of the sphere and no alteration of density 

 will be thereby produced. Conceive to be impressed on the 

 sphere at each instant its own accelerative force in a dii'ec- 

 tion contrary to that in which it takes place, and the same ac- 

 celerative force to be impressed on all points of the fluid in the 

 same direction. The sphere will thus be reduced to rest, and the 

 case of motion will become that of a variable stream impinging 

 on a stationary sphere. The effective accelerative force at any 

 point of the surface of the sphere will be the resultant of the 

 above-mentioned impressed force, and of the force impressed 

 at that point by the sphere in motion. This resultant is in 



the direction of a tangent to the surface, and if — be the 



accelerative force of the sphere at the time t, is equal to 

 d V 



~rj sin 6, the angle 6 being that which has already been de- 

 fined. Now by a known theorem of hydro-dynamics, ifp be 

 the pressure, § the density, and / the effective accelerative 

 force at any point s of a line drawn always in the direction of 

 the motion of the particles through which it passes, 



gas "^ 

 In the instance before us the line 5 is along the surface of 

 the sphere, ds = rdd (r being the radius of the sphere), 

 and if i; = 7K sin b t^ f = ml sin 6 cos b t. Hence by sub- 

 stituting and integrating, 



p = Pi+ mbr cos 6 cos b i. 

 K2 



