[26] PROFESSOR STOKES, ON THE EFFECT OF THE INTERNAL FRICTION 



12. Passing to polar co-ordinates, let r be the radius vector drawn from the origin, 9 the 

 angle which r makes with the axis of x, and let R be the velocity along the radius vector, 

 the velocity perpendicular to the radius vector : then 



x = r cos 9, W = r sin 9, u = R cos 9 - 9 sin 9, q — R sin 9 + 9 cos 9. 

 Making these substitutions in (20), (22), (23), and (25), we obtain 



r sin9(Rrd9-Gdr) = df, ..;... (26) 



d 2 xb { sin 9 d ( 1 d\bA ■ 



^ + ^^U^J =0 < 27 > 



d a \^ 2 sin0 d I 1 d^ 2 \ 1 d\j/ 2 

 ~d?~ + ~r^d9{sti9^9) ~»'~dT = ' ' ' * (28) 



dpm -4-^(f±L rd 0.l^±L dr ) m 



y rsin9\dtdr rdtd9 j yy> 



We must now determine \j/ 1 and \js 2 by means of (27) and (28), combined with the equa- 

 tions of condition. When these functions are known, p will be obtained by integrating the 

 exact differential which forms the right-hand member of (29), and the velocities R, 0, if 

 required, will be got by differentiation, as indicated by equation (26). Formulae deduced 

 from (4) and (5) will then make known the pressure of the fluid on the sphere. 



13. Let £ be the abcissa of the centre of the sphere at any instant. The conditions to 

 be satisfied at the surface of the sphere are that when r = r„ the radius vector of the surface, 

 we have 



R=cos9—, 9=-sin0-^. 



dt dt 



Now r, differs from a by a small quantity of the first order, and since this value of r has 

 to be substituted in functions which are already small quantities of that order, it will be suffi- 

 cient to put r m a. Hence, expressing R and 9 in terms of \|/, we get 



d\ls . dp d4r . . dp 



— = a sin 6r — , — £■ m or sin 9 cos 9 — - , when r = a. . . (30) 



dr dt d9 dt 



When the fluid is unlimited, it will be found that certain arbitrary constants will vanish 

 by the condition that the motion shall not become infinite at an infinite distance in the fluid. 

 When the fluid is confined by an envelope having a radius b, we have the equations of con- 

 dition 



d4f d\ls 



-j- = 0, —5 = 0, when r = b (31) 



dr d9 



14. We must now, in accordance with the plan proposed in Section I., introduce the con- 

 dition that the function \^ shall be composed, so far as the time is concerned, of the circular 

 functions sin nt and cos nt, that is, that it shall be of the form Psin nt + Q cos nt, where P 



