1891.] Fluid Motions in two dimensions. 179 



or out of straight tubes, flow past one or more plane laminas 

 fixed in a finite stream whose boundaries are either free or 

 fixed, or in an infinite stream. In all such cases there is a 

 certain number of bounding stream-lines so that, with the no- 

 tation of Prop. I., there will be a certain region in the z plane 

 within which the motion takes place, bounded partly by given 

 fixed straight lines and partly by unknown free stream-lines, 

 and a certain corresponding region in the w plane bounded by 

 parts of straight lines ^r = const. If the motion and the form 

 of the free stream-lines were known we should know the relation 

 between w and z which effects a conformable representation of 

 the region in the w plane upon the region in the z plane in 

 which the motion takes place. Conversely if this relation can 

 be found we shall know the motion and the form of the free 

 stream-lines. 



This relation can be found indirectly by the aid of the 

 function fl introduced in Prop. I. Along a plane rigid wall 

 the direction of the velocity is given, and along a free stream- 

 line the velocity is constant. Hence along a plane rigid wall 

 the imaginary part of 12 is a given constant, and along the free 

 stream-lines the real part of 12 is a given constant. There will 

 exist a relation between O and z by which could be effected 

 a conformable representation of a certain region in the O plane 

 upon the region of the z plane within which the motion takes 

 place. The region in the O plane is bounded by parts of straight 

 lines parallel to the axis of 12 real, corresponding to the plane 

 rigid walls, and parts of a straight line parallel to the axis of 

 £1 imaginary, corresponding to the free stream- lines. The 

 velocity vanishes at angles of the rigid boundary in the z plane, 

 and at points where stream-lines divide ; the corresponding points 

 of the fl plane lie at oo in the direction of XI real, so that the 

 general form of the boundary in the fl plane is as in the figure, 



where the thick lines correspond to rigid walls and the thin 

 lines to free stream-lines, and there are as many thick lines as 

 there are definite prescribed directions of motion. The inter- 

 sections of thick and thin lines correspond to points where a 

 free stream-line starts out from a rigid boundary. 



VOL. VII. PT. IV. 15 



