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



and when 9 = § 



d (I d d 



die'** d~r' dy' = rd~9' 



du' dR du dR 9 dv' de 

 doe' dr ' dy rd9 r ' dee' dr ' 



whence 



dR m (dR d9 e\ 



P r = p-2 M — , T e =-^(— + - (46) 



dr \rd9 dr rl 



In these formulae, suppose r put equal to a after differentiation. Then P r , T e will be the 

 components in the direction of r, 9 of the pressure of the sphere on the fluid. The resolved 

 part of these in the direction of x is 



P, cos 9 - T e sin 9, 



which is equal and opposite to the component, in the direction of x, of the pressure of the 

 fluid on the sphere. Let F be the whole force of the fluid on the sphere, which will evidently 

 act along the axis of x. Then, observing that 2wa 2 sin9d9 is the area of an elementary 

 annulus of the surface of the sphere, we get 



F=2wa t fi-Pr cos 9+ T e sin9) a sm9d9, .... (47) 



the suffix a denoting that r is supposed to have the value a in the general expressions for 

 P, and T e . 



The expression for F may be greatly simplified, without employing the solution of equa- 

 tions (27), (28), by combining these equations in their original state with the equations of 

 condition (30). We have, in the first place, from (26) 



1 d^ 1 M 



r 4 sm9 d9 r sm 9 dr 



Now the equations (30) make known the values of 4r and ~~, and of their differential 



dr 



coefficients of all orders with respect to 9, when r = a. When the expressions for R and 6 are 



substituted in (46), the result will contain only one term in which the differentiation with 



respect to r rises to the second order. But we get from (21), (27), (28) 



d 2 \f/ sin 9 d t 1 d^\ 1 d\// 2 



~dr* = " ~V d~9 VshTe d~9 J ^^'df' 



and the second of equations (30) gives the value for r = a of the first term in the right-hand 

 member of the equation just written. We obtain from (48) and (30) 



dR\ 



T =0 ' 

 dr) „ 



fdR\ sin0d£ _ /9\ 



\rd~9J a ~ ~~~a~ ~di~ lW, 



fdQ\ _i /rfvM 



\dr)~ ~ ix a sin 9\dt) a ' 



