68, 69.] FLUX ACROSS A CURVE. 67 



function of the position of P. Let ty be the function; more 

 precisely, let ty denote the flux across AP from left to right, as 

 regards an observer placed on the curve, and looking along it from 

 A in the direction of P. Analytically, if I, m be the direction- 

 cosines of the normal (drawn to the right) to any element ds of 

 the curve, we have 



p 



........................ (1). 



C 

 = 



J A 



If the region occupied by the liquid be aperiphractic, ^ is 

 necessarily a single-valued function, but in periphractic regions 

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

 For spaces of two dimensions however periphraxy and multiple- 

 continuity become the same thing, so that the properties of T/T, 

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

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

 Arts. 55, 56, where we have discussed the same question with 

 regard to &amp;lt;/&amp;gt;. 



The cyclic constants of i|r, when the region is periphractic, are 

 the values of the flux across the closed curves forming the several 

 parts of the internal boundaiy. 



A change, say from A to B, of the point from which ^ is 

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

 across a line BA, to the value of i/r ; so that we may, if we please, 

 regard ^r as indeterminate to the extent of an arbitrary constant. 



If P move about in such a manner that the value of i|r does 

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

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

 stream-lines, and ^ is called the stream-function. 



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

 to y, the increment of ty is the flux across PQ from left to right, 

 i.e. d^ = u. PQ, or 



u = f ............... ............... (2). 



dy 



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



