Solid Bodies through Viscous Liquid, 701 



in any manner, so that its velocity at the end of the time t 

 is Y(t). We may evidently obtain the result in this case by 

 writing Y'(r)dr for V, and t — r for t in [12], and integrating 

 with respect to t. We thus get 



(dv\^_ _J_f y'(-r)dr __ 1 f°° 



Wo" ^Wj-.^-T) \A™)Jo ^ 



dt^ 



(13)" 



and since T 3 = — fidv/dx , these formulae solve the problem 

 of finding the reaction in the general case. 



There is another method by which the present problem 

 may be treated, and a comparison leads to a transformation 

 which we shall find useful further on. Starting from the 

 periodic solution (8), we may generalize it by Fourier's 

 theorem. Thus 



(jv) = " f Vn&*y/{inlv)dn . . . (14) 



corresponds to 



V(0=l Y n .e**dn, (15) 



where V„ is an arbitrary function of n. 

 Comparing (13) and (14), we see that 





(16) 



It is easy to verify (16). If we substitute on the right for 

 V'(t) from (15), we get 



-TP-\\ ~7Tt Tl inYne mT an; 



and taking first the integration with respect to t, 



J-.-/C-T ) Jo Vh dtl -V\in)- e ' 



whence (16) follows at once. 



As a particular case of (13), let us suppose that the fluid is 

 at rest and that the plane starts at £ = with a velocity which 

 is uniformly accelerated for a time t x and afterwards remains 



