for Differential Equations of Mathematical Physics. 597 



an indication of this would be expected to be given from the 

 form of the solution in the steady case. 



It will be convenient to throw the equations of motion into 

 the non-dimensional form by the following substitutions : — 



x — L<2?', 



y = V> 



where L is the length of any particular part of the moving- 

 body, and x and y are now merely variable numbers ; 



u = IV, 



* = Vv', 



where U is the steady velocity of one of the boundaries, and 

 u' and v' also variable numbers. Under these circumstances 



<f = JJLyjr'. 



Inserting these in the equations of moiion and in the boundary 

 conditions, we find, omitting dashes, 



where C = UL/v. 



In the case where the body is in motion with uniform 

 velocity in the direction of x there are in addition 







M 



~bx 



oy I 



iiid when the boundary is at rest 



> round the moving body 



(10) 



^=0 



dx 



ay 







•ound body at rest. 



(in 



The integral condition in these two cases becomes 



a» 



2 ( ZT- ds = C f ^-ds round the moving body, (12) 

 V on I B« 



3f 



~dn 



body 



•Jbody 



It is in this non-dimensional form that the equations may 

 .Phil. Mag. S. 6. Vol. 41. No. 244. April 1921. 2 R 



** body 



= round body Jit rest. 



(13) 



