NOTES. 245 



Let us take these in order. 



A. The effects of imperfect fluidity, or viscosity, have been treated 

 of in Chapter IX. The resistance due to this cause is proportional to 

 the velocity of the solid, and, for a body of given shape, to its linear 

 dimensions. This has been proved in the text for the case of the 

 sphere, and it is easy, by the method of dimensions, to extend the 

 result to the general case. Thus if I, L, t, T, u, V, p, P be corre 

 sponding lengths, times, velocities, and pressures in two geometrically 

 similar cases of motion, it appears from equations (14) of Art. 180 that 

 we must have 



p P i*.u fjt,U pu pU 

 I l L** ? ~L r = T ~T 



72 TZ 



Hence -=^, 



and pP : PL&amp;gt;= : ~ = lu : LU. 



t J- 



That is, the resultants of the pressures (normal and tangential) on 

 any corresponding areas are proportional to the products of corresponding 

 lines and velocities. 



It must be remembered however that the investigations of Chap 

 ter IX. proceed on the assumption that the motion is slow. Thus it is 

 very doubtful whether the equations of Art. 180, or the boundary 

 conditions of Art. 181, would be applicable to the motion of the stratum 

 of water in contact with the side of a ship in rapid motion. 



B. It appears from Art. 30 that there is a certain limit to the 

 velocity of a solid of given shape if the motion be continuous. When 

 this limit is reached, the pressure at some point of the surface of the 

 solid sinks to zero, and if the limit be exceeded, a surface of discontinuity 

 is formed. See Art. 94. If the surface of the solid have a sharp 

 projecting edge or angle, this limiting velocity is very small (zero if the 

 edge be of perfect geometrical sharpness), whilst for bodies of fair easy 

 shape it may be considerable. 



When the motion is continuous a certain amount of momentum is 

 expended in each unit of time in starting the elementary streams which 

 diverge from the body in front, but this momentum is restored again to 

 the body by the streams as they close in behind. But when a surface 

 of discontinuity is formed the streams do not close in again ; on the 

 contrary, we have a mass of dead water following the body and 

 pressing on its rear with merely the general pressure which obtains in 



