1891.] 



Fluid Motions in two dimensions. 



197 



where the point marked oo corresponds to the point from which 

 the stream comes and the point marked 1 is the origin in the 

 fl j)lane. 



Now this polygon has right angles at 1, — 1, c, b, an angle 

 at a and an angle lir at oo and it can be conformably represented 

 upon the half-plane u by means of the relation 



d J} = A (30 ) 



In the general case the integration will require elliptic 

 functions and I do not propose to proceed with it. 



13. In the case of symmetry it is clear that there is no 

 flow across the line perpendicular to the lamina at its middle 

 point so that this line may be treated as a real boundary and 

 the a-reo-ion will be 



and the image of this in the line A B. It is clear that the 

 motion takes place as if the boundary were ABODE. If now 

 we reflect this in the line DE we get the same figure as in the 

 escape of a jet from a rectangular vessel by an orifice in the 

 middle of the base. This is a particular case of our first problem 

 and the form of the jet has been worked out by Mr Michell. 

 The relations between z, w, and u, are 



dw 

 du 

 dz 

 du 



l v(i-« 2 ) + V(i-0 



(31), 



J(u 2 -a 2 ) J 



and we propose to find the pressure on the side of the vessel 

 in which the aperture is. If we take the part to the left of 

 the aperture we shall have for the difference of pressure on the 

 two sides of this plane 



J a 



-SX 



dz 



.- du 

 du 



.(32) 



1G— 2 



