246 NOTES. 



the parts of the fluid which are at rest. Hence the restoration of 

 momentum no longer takes place, and a force equal to the loss per unit 

 time must be applied to the solid in order to maintain its motion. 



The only case of this kind which has as yet been mathematically 

 worked out is that in which the -solid is a long plane lamina, with 

 parallel straight edges. If the lamina move broadside-on, the motion is 

 obtained from Art. 98 by considering a velocity q parallel to y impressed 

 on everything. The resistance to the motion of the lamina with velocity 



q n is then - - pq n 2 l, where I is the breadth of the lamina. The resistance 



experienced by a lamina moving obliquely has been calculated by Lord 

 Rayleigh (I. c. Art. 96). The result is 



TT cos a 



where a is the angle which the direction of motion of the lamina makes 

 with the normal to its plane. 



Generally the resistance due to this cause is proportional to the 

 square of the velocity, so long as we can assume that the motions of the 

 fluid corresponding to different velocities of the solid are geometrically 

 similar. 



The formation of the wake which is observed to follow a vessel in 

 motion has been attributed by some writers to the friction between the 

 surface of the ship and the water. It is supposed that the tendency of 

 this friction is to drag a mass of water bodily after the ship. This 

 explanation, if correct, shews that the laws of fluid friction laid down in 

 Chapter IX. do not hold for such rapid motions as are here in question. 



It seems possible however that the wake may be in a great measure 

 due to the cause at present under consideration. It is at all events 

 admitted that the wake is greatly increased by discontinuity in the 

 lines of the vessel, a state of things favourable as we have seen to the 

 formation of a surface of discontinuity in the fluid. Again * it is 

 possible that the bottom of a ship (especially when foul) is to be re 

 garded not as a geometrical surface, but as a mass of projections each of 

 which establishes a surface of discontinuity, with dead water in its rear, 

 on its own account. The aggregate of these masses of dead water would 

 then build up the wake. We should thus also have an explanation of 

 the law, laid down by some writers on this subject, that the skin- 

 resistance is proportional to the square of the velocity. 



* Cf. Stokes, Cavil). I rans., vol. viu, p. 301. 



