73-75] 



DISCONTINUOUS MOTIONS. 



103 



plane of w, to straight lines i/r = constant. There are further 

 conditions of correspondence between special points, one on the 

 boundary, and one in the interior, of each area, which render the 

 problem determinate. These will be specified, so far as is neces 

 sary, as occasion arises. The problem thus presented is a particular 

 case of that solved by Schwarz, in the paper already cited. His 

 method consists in the conformal representation of each area in 

 turn on a half-plane*; we shall find that, in such simple cases as 

 we shall have occasion to consider, this can be effected by the 

 successive use of transformations already studied, and figured, in 

 these pages. 



When the correspondence between the planes of f and w has 

 been established, the connection between z and w is to be found, 

 by integration, from the relation dzjdw = f. The arbitrary con 

 stant which appears in the result is due to the arbitrary position 

 of the origin in the plane of z. 



75. We take first the case of fluid escaping from a large 

 vessel by a straight canal projecting in wards &quot;f. This is the two- 

 dimensional form of Borda s mouthpiece, referred to in Art. 25. 



-I 



A 



-I 



A X 



The figure shews the forms of the boundaries in the planes of 



* See Forsyth, Theory of Functions, c. xx. 



t This problem was first solved by von Helmholtz, /. c. ante p. 24. 



