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



normal (drawn to the left) to any element Ss of the curve, we 

 have 



rP 



= 1 (lu + mv)ds (1). 



J A 



If the region occupied by the liquid be aperiphractic (see p. 43), 

 ty is necessarily a single-valued function, but in periphractic regions 

 the value of ty may depend on the nature of the path AP. 

 For spaces of two dimensions, however, periphraxy and multiple- 

 connectivity become the same thing, so that the properties of ^, 

 when it is a many-valued function, in relation to the nature of 

 the region occupied by the moving liquid, may be inferred from 

 Art. 50, where we have discussed the same question with regard 

 to $. The cyclic constants of ty, when the region is peri 

 phractic, are the values of the flux across the closed curves 

 forming the several parts of the internal boundary. 



A change, say from A to B, of the point from which &amp;gt;/r is 

 reckoned has merely the effect of adding a constant, viz. the flux 

 across a line BA, to the value of ^ ; so that we may, if we 

 please, regard ty as indeterminate to the extent of an additive 

 constant. 



If P move about in such a manner that the value of ^ does 

 not alter, it will trace out a curve such that no fluid anywhere 

 crosses it, i.e. a stream-line. Hence the curves ^ = const, are the 

 stream-lines, and i/r is called the stream-function. 



If P receive an infinitesimal displacement PQ (= %) parallel 

 to y, the increment of A/T is the flux across PQ from right to left, 

 i.e. &\/r = u. PQ, or 



u ** ........................... (2). 



dy 



Again, displacing P parallel to x t we find in the same way 



The existence of a function ty related to u and v in this manner 

 might also have been inferred from the form which the equation of 

 continuity takes in this case, viz. 



J + ^ = () ....... (4), 



dx dy 



