1 C* heir = _ 2\/t 



(17) 



702 Lord Rayleigh o?i iAg Motion of 



constant. Thus from —yo to 0, V(t) = 0; from to n, 

 V(t) = 7it; from t x to £, where £ > t^ V(t)=/it!. Thus 



(0 < * < Tl ) 



and (* > tJ 



/^\ = _L_ f T ' 7 ^ r ?* f / 



. . . (18) 

 Expressions (17), (18), taken negatively and multiplied 

 by P> g ive the force per unit area required to propel the plane 

 against the fluid forces acting upon one side. The force 

 increases until t = r l9 that is so long as the acceleration 

 continues. Afterwards it gradually diminishes to zero. 

 For the differential coefficient of \A~\/(^~~ T i) is negative 

 when t>r 1 ; and when t is great, 



*/t—</(t—Ty) = \T 1 t-i ultimately. 



§ 4. In like manner we may treat any problem in which the 

 motion of the material plane is prescribed. A more difficult 

 question arises when it is the forces propelling the plane that 

 are given. Suppose, for example, that an infinitely thin 

 vertical lamina of superficial density cr begins to fall from 

 rest under the action of gravity when t = 0, the fluid being 

 also initially at rest. By (13) the equation of motion may 

 be written 



rfV 2pv» f « Y'Mdr 



W + ™i\j(t=r)=9,-- ■ ■ ■ (19) 



the fluid being now supposed to act on both sides of the 

 lamina. 



By an ingenious application of AbePs theorem Bogo-io has 

 succeeded in integrating equations which include (19)*. The 

 theorem is as follows: — If ^(0 be defined by 



then 



^mf i=w{m - m . . . (2 i) 



* Bog-gio, Rend. d. Accad. d. Lincei, vol. xvi. pp. 613, 780 (]907) ; also 

 Basset, Quart. Journ. of Mathematics, No. 164. 1910, from which I first 

 became acquainted with Boggio's work. 



f 



