144 REPORT— 1898. 



component vary, as we know, as c^ — z^ , so that if m', v^ denote the mean 

 components along a line perpendicular to the walls 



and (1) and (2) give 



il = _ h'u\ ^ = - ''^J'v^ ^"i' + '^''' = 0. (3) 



dx c^ ' dy c^ ' dx dy 



If 1^ be the stream-line function, taken, say, with reference to the 

 mean velocities it', v^, 



dT^f= u^dy—v^dx, 



and the elimination of p from the first two equations (3) gives 



^ + ^^ = 0. (4) 



dx^ ^ dy"- ^ ' 



The general partial differential equation (4), combined with the condition 

 that the boundaries shall be stream lines, serves to determine completely 

 the function \^. It may be remarked that the lines of equal pressure are 

 the orthogonal trajectories of the stream lines, and can therefore be 

 drawn from the photographs. If we suppose the stream lines equally 

 spaced out in a part of the fluid where the flow is uniform in parallel 

 lines, the velocity at any point will be inversely as the distance between 

 consecutive stream lines. This statement is subject to a qualification 

 -which will be mentioned presently. 



Let us turn now to the other problem, that of determining the stream 

 lines for the irrotational motion in two dimensions of a perfect liquid 

 flowing past an infinitely long body, a transverse section of which, by two 

 close parallel planes, would form the obstacle in our thin plate of highly 

 viscous liquid. In this case the stream line function satisfies the same 

 partial differential equation (4) as before, and the conditions at the 

 boundaries are the same, namely, that the boundaries shall be stream lines. 

 Therefore, notwithstanding the wide difference in the physical conditions, 

 the stream lines are just the same in the two cases. In this latter case 

 they cannot be almost realised experimentally by means of an almost 

 perfect fluid on account of the instability of the motion. The orthogonal 

 trajectories of the stream lines are lines of equal velocity- potential, but 

 not in this case lines of equal pressure. 



It may be objected that the stream lines cannot be the same in the 

 two cases, inasmuch as the perfect liquid glides over the surface of the 

 obstacle, whereas in the case of the viscous liquid the motion vanishes at 

 the surface of the obstacle. This is perfectly true, and forms the qualifi- 

 cation above referred to ; but it does not affect the truth of the propo- 

 sition, which applies only to the limiting case of a viscid liquid confined 

 between walls which are infinitely close. Any finite thickness of the 

 stratum of liquid will entail a departure from the identity of the stream 

 lines in the two cases, which, however, will be sensible only to a distance 

 from the obstacle comparable with the distance between the walls, and 

 therefore capable of being indefinitely reduced by taking the walls closer 

 and closer together. 



