248 SIR WILLIAM THOMSON ON VORTEX MOTION. 



excess of the value of \(f — p, at the end towards which tangential velocity is 

 reckoned as positive, above its value at the other end. 



59. (e). The condition that udx + vdy + w dz is a complete differential [proved 

 above (§ 13) to be the criterion of irrotational motion] means simply 



That the flow [defined § 60 (a)] is the same in all different mutually recon- 

 cilable lines from one to another of any two points in the fluid ; or, which is the 

 same thing, 



That the circulation [§ 60 (a)~\ is zero round every closed curve capable of being 

 contracted to a point without passing out of a portion of the fluid through which the 

 criterion holds. 



From Proposition (1), just proved, we see that this condition holds through 

 all time for any portion of a moving fluid for which it holds at any instant ; and 

 thus we have another proof of Lagrange's celebrated theorem (§ 16), giving us a 

 new view of its dynamical significance, which [see for example § 60 (g)~\ we shall 

 find of much importance in the theory of vortex motion. 



(/). But it is only in a closed curve, capable of being contracted to a point without 

 passing out of space occupied by irrotationally moving fluid, that the circulation 

 is necessarily zero, in irrotational motion. In § 57 we saw that a continuous fluid 

 mass, occupying doubly or multiply continuous space, may move altogether irro- 

 tationally, yet so as to have finite circulation in a closed curve PP'QQ'P, provided 

 PP'Q and PQ/Q, are " irreconcilable paths " between P and Q. That the circula- 

 tion must be the same in all mutually reconcilable closed curves (compare § 57), 

 is an immediate consequence from the now proved [§ 59 (Prop. 2)] equality of 

 the flows [§ 60 (#)] in all mutually reconcilable conterminous arcs. For by 

 leaving one part of a closed curve unchanged, and varying the remaining 

 arc continuously, no change is produced in the flow, in this part; and, by 

 repetitions of the process, a closed curve may be changed to any other recon- 

 cilable with it. 



60. Definitions and elementary propositions (a). The line-integral of the 

 tangential component velocity along any finite line, straight or curved, in a 

 moving fluid, is called the flow in that line. If the line is endless (that is, if 

 it forms a closed curve or polygon), the flow is called circulation. The use of 

 these terms abbreviates the statements of Propositions (2) and (1) of § 59 to the 

 following : — 



[§ 59, Prop. (2)]. The rate of augmentation, per unit of time, of the flow 

 in any terminated line which moves with the fluid, is equal to the excess of the 

 value of ^q" 2 — p at the end from which, above its value at the end towards which, 

 positive flow is reckoned. 



[§ 59, Prop. (1)]. The circulation in any closed line moving with the fluid, 

 remains constant through all time. 



(b). If any open finite surface, lying altogether within a fluid, be cut into 



