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



tions HI 2 (k — l) and tn 2 &' remain nearly constant. Hence for a considerable range of values 

 of TO, each of the functions may be expressed pretty accurately by J + Bm. When HI 

 is at all large, the first two terms in the 2nd and 3rd of the formulae (113) will give k and k' 

 with considerable accuracy, because, independently of the decrease of the successive quan- 

 tities tn~', HI -2 , III" 3 ..., the coefficients of III -1 and III" 2 are considerably larger than those 

 of several of the succeeding powers. If we neglect in these formulas the terms after « 2 , 

 we get 



k = i + ^2. m-\ k'= -v/2. tn- 1 + ^ m- 2 . 



It may be remarked that these approximate expressions, regarded as functions of the radius 

 a, have precisely the same form as the exact expressions obtained for a sphere, the coefficients 

 only being different. 



Section IV. 



Determination of the motion of a fluid about a sphere which moves uniformly with 

 a small velocity. Justification of the application of the solutions obtained in Sections II. 

 and III. to cases in which the extent of oscillation is not small in comparison with the 

 radius of the sphere or cylinder. Discussion of a difficulty which presents itself with 

 reference to the uniform motion of a cylinder in a fluid. 



39. Let a sphere move in a fluid with a uniform velocity V, its centre moving in a right 

 line ; and let the rest of the notation be the same as in Section II. Conceive a velocity 

 equal and opposite to that of the sphere impressed both on the sphere and on the fluid, which 

 will not affect the relative motion of the sphere and fluid, and will reduce the determination 

 of the motion of the fluid to a problem of steady motion. Then we have for the equations 

 of condition 



R = o, 9 = 0, when r = a; (116) 



B=-Vcosd, 6 = Fsinfl, when r= oo (117) 



The equations of condition, as well as the equations of motion, may be satisfied by sup- 

 posing \j/ to have the form sin 2 Of (r). We get from (20'), by the same process as that by 

 which (33), (34) were obtained, 



(&-S)V»-« (,18) 



the only difference being that in the present case the equation (20') cannot be replaced by the 

 two (22), (23), which become identical, inasmuch as the velocity of the fluid is independent 

 of the time. 



The integral of (118) is 



f{r) = Ar~ x + Br + CV + Dr\ (119) 



