Solid Bodies through Viscous Liquid. 703 



For by (20), if (*-t)»=», 



•^ 



so that 



(*-t)* ~~jo 



f'st-j>-*->- 



= 4^ clA $>(t-x*-y*)dy 



»/ o Jo 



Jo 



where r 2 =x 2 +y 2 . 



Now, if £' be any time between and t, we have, as in (19), 



( }+ ^Oo X/C^-T) 



#■ 



Multiplying this by (t — t') hit' and integrating between 

 and t, we get 



f ' Y'(t')dif 2 P i*C* dt' C* N r {T)dT _ C* dt' 



Jo (t-t'f an*} (t-tfj (^-t)* ^Jo (*-f)*' 



(22) 



In (22) the first integral is the same as the integral in (19). 

 By Abel's theorem the double integral in (22) is equal to 

 ttV(0, since V(0) = 0- Thus 



I£ we now eliminate the integral between (19) and (23) 3 

 we obtain simply 



dV 4p 2 ./ ir <kpvi 



---yr-g-J^gJt . . . (24) 



as the differential equation governing the motion of the 

 lamina. 



This is a linear equation of the first order. Since V 



