﻿of a Fluid to a Plane. 411 



found by the following method, with result expressed in § 9, 

 which 1 venture to give as a guess, and not as a satisfactory 

 mathematical investigation. 



§ 5. Considering a disk of finite thickness, however small, 

 moving in an inviscid incompressible liquid within an un- 

 yielding boundary, and, for a moment, thinking only of the 

 ?f-component of the motion, according to the notation of § 1 , 

 let E and E' denote the front and the rear parts of the edge, 

 respectively. Imagine now instead of the real motion of the 

 unvarying solid disk through the fluid, that the disk grows 

 all over E, by rigidification and accretion of the fluid in front 

 of it, and melts away from E' by liquefaction of the solid. 

 In an infinitesimal time 67, the extent of the accretion in front 

 of E will be u$t. Now if the v component of the motion of 

 the disk is maintained without diminution during this accre- 

 tion, a force, F, equal to (!' — I)/Bt, must be applied from 

 without, perpendicular to the disk ; I denoting the impulsive 

 force which would be required to give the ^-component 

 velocity to the unaugmented disk, and V that required to give 

 the same velocity to the augmented disk. The point of 

 application of the force (V — T)/Bt must be that of the 

 resultant of impulses I' and —I, applied at the hydraulic 

 centres of inertia* of the augmented disk and the unaug- 

 mented disk respectively. 



§ 6. Sudden cessation of the rigidity by liquefaction of any 

 portion, (finite or infinitely small) of matter of the disk at E' 

 requires no instantaneous application of force, to prevent 

 change of the v-motion of the residual solid. The continued 

 gradual liquefaction which we are supposing performed, leaves 

 a Helmholtz " vortex sheet " of finite slip growing out in the 

 liquid, behind E', the evolutions and contortions of which are 

 not easily followed in imagination. This sheet is in the form 

 of a pocket of which the lip remains always attached to the 

 solid disk. The space enclosed between it and the disk is 

 filled by the liquid which was solid. It grows always longer 

 and longer by gain of liquid from the melting solid at E' in 

 front of it, and probably also by its rear extending farther 

 and farther, far away in the wake of the disk. 



§ 7. Suppose now that, after having been performed during 

 a certain time T, the ideal processes of §§ 5, 6 are discon- 

 tinued, and the resulting solid disk, equal and similar to the 

 original disk, but carried in the ^-direction through a space 



* I call the " hydraulic centre of inertia " of a massless rigid disk im- 

 mersed in liquid the point at which it must be struck perpendicularly 

 by an impulse, to give it a simple translational motion. 



