69-71] INDIRECT METHODS. 91 



been obtained by an inverse process ; viz. instead of trying to find 

 a solution of the equation V 2 &amp;lt; = or V 2 -\Jr = 0, satisfying given 

 boundary conditions, we take some known solution of the differen 

 tial equations and enquire what boundary conditions it can be 

 made to satisfy. Examples of this method have already been 

 given in Arts. 63, 64, and we may carry it further in the following 

 two important cases of the general problem in two dimensions. 



71. CASE I. The boundary of the fluid consists of a rigid 

 cylindrical surface which is in motion with velocity u in a 

 direction perpendicular to its length. 



Let us take as axis of x the direction of this velocity u, and 

 let Bs be an element of the section of the surface by the plane xy. 



Then at all points of this section the velocity of the fluid in 

 the direction of the normal, which is denoted by d-fr/ds, must be 

 equal to the velocity of the boundary normal to itself, or udyjds. 

 Integrating along the section, we have 



T/T = My + const (1). 



If we take any admissible form of i/r, this equation defines a 

 system of curves each of which would by its motion parallel to 

 x give rise to the stream-lines ^r = const. * We give a few 

 examples. 



1. If we choose for ^r the form uy, (1) is satisfied 

 identically for all forms of the boundary. Hence the fluid 

 contained within a cylinder of any shape which has a motion 

 of translation only may move as a solid body. If, further, the 

 cylindrical space occupied by the fluid be simply-connected, this is 

 the only kind of motion possible. This is otherwise evident from 

 Art. 40 ; for the motion of the fluid and the solid as one mass 

 evidently satisfies all the conditions, and is therefore the only 

 solution which the problem admits of. 



2. Let -v/r = A jr . sin 6 ; then (1) becomes 



A 



sin 6 = - u r sin 6 + const (2). 



In this system of curves is included a circle of radius a, provided 



* Cf. Kankine, &quot;On Plane Water-Lines in Two Dimensions,&quot; Phil. Trans., 

 1864, where the method is applied to obtain curves resembling the lines of ships. 



