50 IRROTATIONAL MOTION. [CHAP. III. 



irrotationally. For the two circuits may be connected by a con 

 tinuous surface lying wholly within the region ; and if we apply 

 the theorem of Art. 40 to this surface, we have, remembering the 

 rule as to the direction of integration round the boundary, 



I (ABC A) + 1 (AC B A} = 0, 

 or / (ABGA) = I (A B C A). 



If a circuit ABC A be reconcileable with two or more circuits 

 A B C A , A&quot;B&quot;C&quot; A&quot;, &c., combined, we can connect all these cir 

 cuits by a continuous surface which lies wholly within the region, 

 and of which they form the complete boundary. Hence 



I (ABC A) + 1 (A C B A] + I(A&quot;C&quot;B&quot;A&quot;) + &c. = 0, 

 or I (ABC A) = I(A B C A) + I(A&quot;B&quot;C&quot;A&quot;) + &c. ; 



i. e. the circulation in any circuit is equal to the sum of the cir 

 culations 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 non-evanescible circuits a j) a^,...a n can be 

 drawn in it ; and let the circulations in these be tc lt vc 2 , . . . /c n , respect 

 ively. 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 a lt a 2 ,...a w ; say with a x taken ^ 

 times, a 2 taken^&amp;gt; 2 times and so on, where of course any^&amp;gt; is nega 

 tive when the corresponding circuit is taken in the negative 

 direction. The required circulation then is 



Since any two paths joining two points A, B &quot;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 (14), where, of 

 course, in particular cases some or all of the j&amp;gt; s may be zero. 



