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



Eliminating p by differentiation from the two equations (66), and expressing u and t> in 

 terms of ^ m tne resulting equation, we get 



/ d 2 d 8 • 1 d \ l <P d?_\ 



\d^ + d^~^dtj\d^ + 'dy) x ~ ' ' ' ' • 



and, as before 

 where 



X = Xi + X" • 



f d" d* \ 



{c^ + d?)*^ ' 



f <F d 2 1_ d\ 



K2& + dtf " n' dt) X * ~ ° 



. . . (69) 



. , . (70) 



• • • (71) 



. . . (72) 



We get from (66) and (68) 



, d/d 2 d 2 1 d \ , d / d 2 d 2 1 d \ 



dp = »pdx.-\^ + ^--,-) x -»pdy.-{— + —--,-) x , 



which becomes by means of (70), (71), and (72) 



26. Passing to polar co-ordinates r, (9, where 9 is supposed to be measured from the axis 

 of x, we get from (68), (71), (72), and (73) 



RrdO- 9dr = d x , 



id Id 2 



1 s id 1 <P \ 



/£_ l _d 



ydr 1 r dr 



(74) 

 (75) 



Id l 



r r* 



1 d\ 



(76) 

 (77) 



R, in (74) being the velocities along and perpendicular to the radius vector. 



27. Let a be the radius of the cylinder ; and as before let the cylinder's motion be 

 defined by the equation 



dt 



" = ce* 7 * '; 



(78) 



then we have for the equations of condition which relate to the surface of the cylinder 



R = —£r = COS0 -2 = C COS0 e' x '' n^ ^ 



rd0 dt 



dv d? 



= - -7* = - sin $ -5 = - c sin (9 e"' m ''. 

 dr d* 



when r = a. 



(79) 



