of Attractive Forces. 327 



the velocity and condensation in a plane through the centre of 

 the sphere perpendicular to the direction of the propagation of 

 the waves, we shall have 



For reasons already given, the velocity which the reaction of the 

 surface of the sphere impresses on the incident fluid will be 

 W cos 6, and may be assumed without sensible error to be im- 

 pressed as if the fluid were incompressible. Hence the velocity 

 along the surface (V) is equal to W sin 6, and the usual equation 

 gives 



JdW V^ 



-^s,me.cdd+-^=f{t). 



IT 



Determining the arbitrary function so that p=Pi where Q-= -^, 

 it will be found that 



«VNap.log^ = ^ccos6'+-2-cos2^. . . . (^) 



Since there is no reaction of the sphere at points corresponding 



to 6=—, pi is the density of the incident wave, and is therefore 



equal to l+a. Hence, putting l + s for p the density of the 

 fluid in contact with the sphere, and having regard to the equa- 

 tion W = Kaa; the following result is obtained : — 



«V(.-c.) = (-^+-.^>cos^ + i(W^ + ^,.-^Jcos^^. 



This equation gives the pressure on that half of the spherical 

 surface on which the waves are incident. T proceed next to in- 

 vestigate an expression for the pressure on the other half. 



First, it may be remarked that if terms of the second order be 

 omitted, the equations (a) and (|S) are identical, excepting that 



the latter has — in the place of p. Hence it may be inferred 



Pi . . , 



that to this order of approximation the same pressure on the 



sphere is produced, whether the sphere move in fluid of a certain 

 density at rest, or fluid of the same density be incident on the 

 sphere at rest, the motions of the fluid and the sphere being in 

 the two cases the same. This result is also deducible as follows. 

 The pressure on the surface of a moving sphere having been de- 

 termined by the solution of Problem TIL, conceive an equal and 

 opposite motion to be impressed upon it and upon all the parts of 

 the fluid at each instant. The sphere is thus reduced to rest, and 

 the fluid is made to impinge upon it. But no dynamical action 

 is produced between the sphere and the fluid by impressing the 

 same velocities on both. Hence the pressure on the sphere is 



