1911] ON THE MOTION OF SOLID BODIES THROUGH VISCOUS LIQUID 33 As a particular case of (13), let us suppose that the fluid is at rest and that the plane starts at t = 0 with a velocity which is uniformly accelerated for a time TZ and afterwards remains constant. Thus from — oo to 0, F(r) = 0; from 0 to TI} F(r) = /ir; from TI to t, where t>r1} F(r) = Ar1. Thus (0 < t < TI) ,dv\___1 ft hdr %h\/t \dasj a ^(-TTV) Jo V(* — T) and (t > TJ) dv l Tl h dr .(17) Expressions (17), (18), taken negatively and multiplied by p, give the force per unit area required to propel the plane against the fluid forces acting upon one side. The force increases until t=r1} that is so long as the acceleration continues. Afterwards it gradually diminishes to zero. For the differential coefficient of \/t — *J(t — TI) is negative when t > TI ; and when t is great, V* — \/(t — TI) = ^rlt~ * 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 the fluid being now supposed to act on both sides of the lamina. By an ingenious application of Abel's theorem Boggio has succeeded in integrating equations which include (19)*. The theorem is as follows: — If ty(t) be defined by then J^w."=Tr{<l>(t)-$(()}}......................(21) (t - T)* ' (t-y')dy, o * Boggio, ttend. d. Accad. d. Lincei, Vol. xvi. pp. 613, 730 (1907); also Basset, Quart. Journ. of Mathematics, No. 164, 1910, from which. I first became acquainted with Boggio's work. K. VI. 3blem of § 7 : instead of the motion of the 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 F.. The stream-function (ty) for this motion satisfies the same differential equation as does the transverse displacement (w'} of a plane elastic plate. And a surface on which the fiuid remains at rest (i/r = 0, d^/dn = 0) corresponds to a curve along which the elastic plate is clamped.