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



Since the second member of this equation must be independent of t, we get y (#') = 

 a constant, and this constant must be equal to V, since 



e" s ds= 1. 



Substituting in (185) we get 



V /•» (*+*')' 

 ■o = ■ — — / e in't dw (186) 



V7TM ' 



For the object of the present investigation nothing is required but the value of — for m = 0, 



doo 



which we may denote by ( — ) . We get from (186) 



\dool 



/dv\ V 



( — ) = 7= (187) 



\dx/o y/ tt fit 



Now suppose the plane to be moved in any manner, so that its velocity at the end of the 

 time t is equal to f(t). We may evidently obtain the result for this case by writing 

 /' (t') dt' for V, and t - t' for t in (187), and integrating with respect to t' . We thus get 



A 1 /•«-,, ,v dt' 1 /■« . , , dt. 



dB 

 To apply this result to the case of an oscillating disk, let r — = rF(t) be the velocity of 



any annulus, and G the moment of the whole force of the fluid on the disk. Then 



fdv, 



G-**»'pf m *(P) dr; 



J \da!jo 



an d (—\ w iH be got from (188) by substituting rF{t) for f{t). We find thus 

 \dxj 



dt t 



G=-^Tr»'.pa* [ F'(t-tO-± (1 



89) 



If we suppose the angular velocity of the disk to be expressed by A sin nt, where A is 

 constant, we must put F(t) = A sin nt in (189), and we should then get after integration the 

 same expression for G as was obtained in Art. 8 by a much simpler process. Suppose, 

 however, that previously to the epoch from which t is measured the disk was at rest, and 

 that the subsequent angular velocity is expressed by A t sin nt, where A t is a slowly varying 

 function of t. Then 



.F(£)=0 when t<0, F(t) = A, sin nt when t>0. 



On substituting in (189) we get 



dt x 

 V~t 



G= - \Arfx'.pa*n f A t _ h cos n(t - «,) — ~. . . . (190) 



