256 SIR WILLIAM THOMSON ON VORTEX MOTION, 



and the preceding becomes 



di -" ^ 



But if the tangential component velocities at P x and P 2 are not only equal, but 

 in the same direction, q> z — <p L must, as we have seen, be constant over the 

 membrane, and therefore w must also be constant. 



Suppose now that after pressure has been applied for any time in the manner 

 described, of uniform value all over the membrane at each instant, it is applied 

 no longer, and the membrane (having no longer any influence) is done away 

 with. The fluid mass is left for ever after in a state of motion, which is irrota- 

 tional throughout, but cyclic. The " circulation" [§ 60 («)], or the cyclic constant 

 being equal to ?., — <p v for every circuit reconcilable with P 1 PP 2 P 1 is given by the 

 equation 



<P- 2 - ?x — " — fait (4), 



fdt denoting a time-integral extended through the whole period during which ©• 

 had any finite value. 



The same kind of operation may be performed, on each of the n barriers 

 temporarily introduced in § 61 to reduce the (w + l)fold continuity of the space 

 occupied by the fluid, to simple continuity. 



The velocity potential at any point of the fluid will then be a polycyclic func- 

 tion [§ 60 {x)~\ equal to the sum of the separate values corresponding to the 

 pressure separately applied to the several barriers. Thus we see how a state of 

 irrotational motion, cyclic with reference to every one of the different circuits of 

 a multiply continuous space, and having arbitrary values for the corresponding 

 cyclic constants, or circulations, may be generated. But the proof of the 

 possibility of fluid motion fulfilling such conditions, founded on this planning out 

 of a genesis of it, leaves us to imagine that it might be different according to the 

 infinitely varied choice we may make of surfaces for the initial forms of the 

 barriers, or according to the order and the duration of the applications of 

 pressure to them in virtue of which these figures may be changed more or less, 

 and in various ways, before the initiating pressures all cease; and hitherto 

 we have seen no reason even to suspect the following proposition to the con- 

 trary. 



63. (Prop.) The motion of a liquid moving irrotationally within an (w+l)ply 

 continuous space is determinate when the normal velocity at every point of the 

 boundary, and the values of the circulations in the n circuits, are given. 



This is proved by an application of Green's extended formula (7) of § 57, 

 showing, as the simple formula (1) of the same section showed us in § 61 for 

 simply continuous space, that the difference of the velocity potentials of two 

 motions, each fulfilling this condition, is necessarily zero throughout the whole 



