56 . IRROTATIONAL MOTION. [CHAP. Ill 



circulations in the several members of any set of circuits with 

 which it is reconcileable. 



Let the order of connection of the region be n + 1, so that 

 n independent simple irreducible circuits Oi, a 2) ...a n can be 

 drawn in it; and let the circulations in these be K I} K 2 ,.../c n , 

 respectively. The sign of any K will of course depend on the 

 direction of integration round the corresponding circuit ; let the 

 direction in which K is estimated be called the positive direction 

 in the circuit. The value of the circulation in any other circuit 

 can now be found at once. For the given circuit is necessarily 

 reconcileable with some combination of the circuits 1} a 2 ,...a w ; 

 say with a^ taken p l times, a 2 taken p. 2 times and so on, where of 

 course any p is negative when the corresponding circuit is taken 

 in the negative direction. The required circulation then is 



p lKl +p 2 tc 2 +... +p n Kn (1). 



Since any two paths joining two points A, B of the region 

 together form a circuit, it follows that "the values of the flow in 

 the two paths differ by a quantity of the form (1), where, of 

 course, in particular cases some or all of the p's may be zero. 



50. Let us denote by <f> the flow to a variable point P from a 



fixed point A, viz. 



rp 

 </> = - I (udx+vdy+wdz) (2). 



^i 



So long as the path of integration from A to P is not specified, 

 </> is indeterminate to the extent of a quantity of the form (1). 



If however n barriers be drawn in the manner explained in 

 Art. 48, so as to reduce the region to a simply-connected one, 

 and if the path of integration in (2) be restricted to lie within 

 the region as thus modified (i.e. it is not to cross any of the 

 barriers), then <j> becomes a single-valued function, as in Art. 36. 

 It is continuous throughout the modified region, but its values 

 at two adjacent points on opposite sides of a barrier differ by 

 K. To derive the value of $ when the integration is taken along 

 any path in the unmodified region we must subtract the quantity 

 (1), where any p denotes the number of times this path crosses 

 the corresponding barrier. A crossing in the positive direction of 

 the circuits interrupted by the barrier is here counted as positive, 

 a crossing in the opposite direction as negative. 



