596 Messrs. Cowley an d Levy: Method of Analysis suitable 



boundaries ; then the terms on the left-hand side vanish, 

 and the equation (7) becomes 



rg*. : .... (7) 



* 



Any such contour is reconcilable with a contour round the 

 actual outline of the boundary. Equation (7) has three 

 particular forms, when no external forces exist, of interest 

 to the present problem. Round a body at rest, where 



u, c, =£-, etc. are zero, it takes the form 

 ot 



2v'^ds = (7a) 



- body on 



Round a body moving with constant velocity U in the 

 direction #, 



(2v\Zd*={ ^ (Vv-l^)ds. . . (7b) 



Along a boundary at rest, such as a wall of a channel where 

 the fluid is under a given pressure-head P between the ends 

 maintaining the flow, it becomes 



1 oZ 7 P 



^-ds = - (7 c) 



&n p ' 



v boundaiy 



Before a solution of equation (6) can be accepted, therefore, 

 where the expression for yjr satisfies the appropriate boundary 

 condition of the problem, it is necessary to satisfy the various 

 forms of (7) where the integral is taken round each boundary 

 of the fluid. The boundary conditions that are to be inserted 

 are of course those that involve the statement that there is 

 no slip. 



§ 3. Steady motion in two dimensions. — Consider the case 

 where the boundaries of the fluid are in steady motion, and 

 where in addition it is assumed that the fluid everywhere 

 moves steadily. As far as experimental evidence shows, 

 these two assumptions would appear to be entirely independent 

 of each other; for in all known cases above a certain value 

 of vl/v a steadily moving body appears to give rise to a periodic 

 eddying in its wake. Below this critical value, however, these 

 two assumptions appear to be perfectly consistent. If steady 

 motion of the fluid is not possible above this critical value, 



