700 Lord Rayleigh on the Motion of 



quantities 



v=Y n e- x ^ n ^cos{nt--x</(nfiv)} . . . (6) 



corresponds with 



Y=Y n cosnt (7) 



for the plane itself. 



In order to find the tangential force ( — T 3 ) exercised upon 

 the plane, we have from (5) when # = 



VdxJo 



■YJ»' V '(in/v), .... (8) 



and T s = -p(dv/dx) = P Y 7 ,ft,/(iny') 



= Px /(inv).(l + i)Y n e<>« 



=Pv /(i H .(v + if), .... (9) 



giving the force per unit area due to the reaction of the fluid 

 upon one side. " The force expressed by the first of these 

 terms tends to diminish the amplitude of the oscillations 

 of the plane. The force expressed by the second has the 

 same effect as increasing the inertia of the plane.'"' It will 

 be observed that if V n be given, the force dimin^hes without 

 limit with n. 



In note B Stokes resumes the problem of § 7: instead of 

 the motion of ihe plane being periodic, he supposes that the 

 plane and fluid are initially at rest, and that the plane is 

 then (t = 0) moved with a constant velocity V. This problem 

 depends upon one of Fourier's solutions which is easily 

 verified *. We have 



dx s/(™t) ' ^ 



v/tt 



For the reaction on the plane we require only the value of 

 dv/dx when x = Q. And 



Stokes continues f " now suppose the plane to be moved 



* Compare Kelvin, Ed. Trans. 1862 ; Thomson & Tait, Appendix D. 

 t I have made some small changes of notation. 



