33-35] CIRCULATION. 39 



Integrating along the line, from A to B, we get 



~r\ rB r dr) ~~\ -B 



-jr- I (udx + vdij + wdz) = I O + ^q 2 \ (1), 



UtJA L -&amp;gt; P _U 



or, the rate at which the flow from A to B is increasing is equal 

 to the excess of the value which fdp/p l + ^q 2 has at B over 

 that which it has at A. This theorem comprehends the whole 

 of the dynamics of a perfect fluid. For instance, equations (2) of 

 Art. 15 may be derived from it by taking as the line AB the in 

 finitely short line whose projections were originally 8a, 86, Sc, 

 and equating separately to zero the coefficients of these in 

 finitesimals. 



If H be single-valued, the expression within brackets on the 

 right-hand side of (1) is a single-valued function of a, y, z. 

 Hence if the integration on the left-hand be taken round a closed 

 curve, so that B coincides with A, we have 



gr I (udx + vdy + wdz) = (2), 



or, the circulation in any circuit moving with the fluid does not 

 alter with the time. 



It follows that if the motion of any portion of a fluid mass be 

 initially irrotational it will always retain this property ; for other 

 wise the circulation in every infinitely small circuit would not 

 continue to be zero, as it is initially, by virtue of Art. 33 (4). 



35. Considering now any region occupied by irrotationally- 

 moving fluid, we see from Art. 33 (4) that the circulation is zero 

 in every circuit which can be filled up by a continuous surface 

 lying wholly in the region, or which is in other words capable of 

 being contracted to a point without passing out of the region. 

 Such a circuit is said to be reducible. 



Again, let us consider two paths ACB, ADB, connecting two 

 points A, B of the region, and such that either may by con 

 tinuous variation be made to coincide with the other, without ever 

 passing out of the region. Such paths are called mutually 

 reconcileable. Since the circuit AGED A is reducible, we have 

 I (ACB DA) = 0, or since I(BDA) = - 1 (ADB), 



i.e. the flow is the same along any two reconcileable paths. 



