344 PROFESSOR CHALLIS, ON THE MOTION OF A SMALL SPHERE 



Now since / sin 6 cos 2 dO = -, and I sin 9 cos* 0d9 = 0, the whole 



resulting pressure on the sphere is 



47r«mr 8 i. ... 7 -£l 

 3 VMO -** J- 



And if 5 be the ratio of the density of the fluid to that of the sphere, the 

 accelerative force of the resistance of the fluid is 



-Si 



which on account of the very small factor e r will after a short interval 

 become, 



.—■+«>• 



6. Suppose, for example, the sphere to vibrate as a pendulum, and the 

 extent of the vibrations to be so small that the motion of the centre may 

 be considered rectilinear. Let / be the length of the pendulum, and x the 

 distance of the centre of the sphere at the time t from the position it would 

 have at rest. Then, taking the buoyancy of the fluid into account, we 



have for the accelerative force of gravity — — - (1 - $)•, and consequently, 



by the foregoing reasoning, 



d 2 x gx . «, amS -ili r zl d*x .. ,1 



df I ' r \J mdf* J 



_ T r a 4d*x At Zfr d*x r % d'x\ . _ , , 



Now J e w dt = e {a-dr-a'-d?) yer y nearl y* Hence ' b ^ sub - 



stituting, 



at 



d*x _gx 1 — $ rS d 3 x kam§ 



df ~ T'TTS + a{l+8)"3F + r(\+$) e ' { >' 



Hence, for a first approximation, after a very small time, 



d 2 x ffx /1 — S 



d'x _ gx /l — 6\ 

 df~~~T' [l+li 



This equation not containing a is true of an incompressible fluid. It 

 is, in fact, when applied to this case, an exact equation, as appears 



