90-91] MOTION OF A SPHERE. 131 



of the solid, the amount of the increase being now half the mass 

 of the fluid displaced*. 



Thus in the case of rectilinear motion of the sphere, if no 

 external forces act on the fluid, the resultant pressure is equiva 

 lent to a force 



in the direction of motion, vanishing when u is constant. Hence 

 if the sphere be set in motion and left to itself, it will continue to 

 move in a straight line with constant velocity. 



The behaviour of a solid projected in an actual fluid is of 

 course quite different ; a continual application of force is necessary 

 to maintain the motion, and if this be not supplied the solid is 

 gradually brought to rest. It must be remembered however, in 

 making this comparison, that in a perfect fluid there is no 

 dissipation of energy, and that if, further, the fluid be incompres 

 sible, the solid cannot lose its kinetic energy by transfer to the 

 fluid, since, as we have seen in Chapter in., the motion of the 

 fluid is entirely determined by that of the solid, and therefore 

 ceases with it. 



If we wish to verify the preceding results by direct calculation from the 

 formula 



we must remember, as in Art. 68, that the origin is in motion, and that the 

 values of r and 6 for a fixed point of space are therefore increasing at the 

 rates - u cos 6, and u sin 0/r t respectively. We thus find, for r = a, 



(ii). 



The last three terms are the same for surface-elements in the positions 6 and 

 TT- 6 ; so that, when u is constant, the pressures on the various elements of the 

 anterior half of the sphere are balanced by equal pressures on the correspond 

 ing elements of the posterior half. But when the motion of the sphere is 

 being accelerated there is an excess of pressure on the anterior, and a defect of 

 pressure on the posterior half. The reverse holds when the motion is being 

 retarded. The resultant effect in the direction of motion is 



%7ra sin 6 . ad6 . p cos 0, 

 o 



which is readily found to be equal to - ^npa 3 du/dt, as before. 



* Green, &quot;On the Vibration of Pendulums in Fluid Media,&quot; Edin. Trans., 

 1833 ; Math. Papers, p. 322. Stokes, I c. 



92 



