﻿412 On the Resistance of a Fluid to a Plane. 



equal to wT, is left with simply its v-motion through the fluid 

 maintained. The pocket of liquefied solid will be left farther 

 and farther behind the disk. Its mouth, still always stopped 

 by the solid, will shrink from its original area which was the 

 whole of E' ; and will become always smaller and smaller, but 

 not infinitely small in any finite time. The neck of the pocket 

 in the wake of the disk will become narrower and narrower, 

 and the whole pocket will be drawn out longer and longer 

 behind ; but, through all time, the fluid which was solid will 

 remain separated by a surface of finite slip, or Helmholtz 

 a vortex sheet," from the surrounding fluid, except over the 

 ever diminishing area of the disk, which stops the mouth of 

 the pocket. The motion of the fluid is irrotational outside the 

 pocket, and rotational within it. To keep the solid disk 

 moving with its v-motion constant, and with no other motion 

 whether rotational or translational, it is necessary to apply 

 force to it. But this force becomes less and less, and approxi- 

 mates to zero, as the vortex-trail becomes finer and finer ; 

 and the motion of the fluid in the neighbourhood of the disk 

 approximates more and more nearly to perfect agreement with 

 the unique irrotational motion due to v-motion of the solid 

 through the fluid. 



§ 8. So far we have, in §§ 5, 6, 7, been on sure ground, 

 and every statement is rigourously true, not only for a 

 " disk " of any shape of boundary and of any thickness how- 

 ever small, but also for a solid of any shape, dealt with 

 according to § 5, provided only that the fluid is inviscid and 

 incompressible, and its boundary unyielding. My hypothesis, 

 or " guess" (§ 4), which forms the subject of the present 

 paper, is that default from infinitely perfect fulfilment of all 

 these three conditions would, for an infinitely thin disk kept 

 moving with uniform translational motion (u, v, § 1), require 

 the continued application to it of force determined in magni- 

 tude and position by § 5; provided v be very small in com- 

 parison ivith u. 



§ 9. The result is worked out with great ease for the case 

 of a rectangular disk of which the length, I, is very great in 

 comparison with the breadth, a. For this case, by the well- 

 known hydrokinetics of an ellipsoid or elliptic cylinder 

 moving translationally in an inviscid incompressible fluid of 

 unit density, we have j_.j L7ra 2^ . 



and, still using the notation of § 5_, 



l' = \Tr{a + uht)Hv. 

 Hence F = \iraluv ; 



