98 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV. 



on account of the symmetry we need only consider the motion 

 between these two lines. If the velocity in the canal at a distance 

 from the mouth be taken to be unity, the half-breadth of the 

 canal will be TT. The boundaries in the plane of f are shewn in 

 the figure, corresponding points in the planes of z and f being 

 indicated by corresponding Roman and Greek letters. The point 

 A corresponds to the origin of f because the velocity there is 

 infinite. (Art. SO.) 



We have now to connect f and w by a relation such that 72/3 

 and /3oo shall correspond to the two straight lines i|r = 0, ^ = TT. 

 If we assume z f 2 , then in the plane of z the lines 72/1? and /3oo 

 become parts of the same straight line. The assumption z = 1 + e w 

 then converts these two parts into the straight lines i|r= 0, ^ = TT. 

 See Art. 78. We have then 



whence 



The constant of integration is so chosen that the origin of z 

 (hitherto arbitrary) shall be at the intersection of the middle line 

 of the canal with the plane of its mouth. 



Discontinuous Motions*. 



94. We have had frequent occasion to remark, concerning 

 forms of fluid motion which we have obtained, that they cannot 

 be realized in practice on account of the infinite velocity and 

 consequent negative pressure which they would involve at some 

 point of the boundary. We are led to solutions of this nugatory 

 character whenever a sharp projecting edge forms part of the 

 boundary. Edges of absolute geometrical sharpness do not of 

 course occur in practice ; but even if the edge be slightly rounded, 

 (as for instance in Example 8 above, by the substitution of a 

 neighbouring stream-line as the fixed boundary,) the velocity in 

 the immediate neighbourhood will, unless the motion be every- 



* Helmholtz, I.e. Art. 92. 



