SIR WILLIAM THOMSON ON VORTEX MOTION. 255 



rotational motion through fluid pressure from the motion of its boundary ; and 

 we go on, by aid of Green's extended formula [§ 57 (7)], to prove the determinate- 

 ness of the interior motion under conditions now to be specified for multiply 

 continuous space, as we have done by his unaltered formula [§ 57 (1)] for simply 

 continuous space. 



62. Genesis of Cyclic Irrotational Motion. — In the case of motion considered 

 in § 61, the value of the normal component velocity is not independently arbitrary 

 over the whole boundary, but has equal arbitrary values, positive and negative, 

 on the two sides of each of the barriers /3 1? /3 2 , &c. We must now introduce a 

 fresh restriction in order that, when the barriers are liquefied, the motion of the 

 fluid may be irrotational throughout the space thus re-opened into multiple 

 continuity. For although we have secured that the normal component velocity 

 is equal everywhere on the two sides of each barrier, we have hitherto left the 

 tangential velocity, unheeded. If they are not equal on the two sides, and in 

 the same direction, there will be a finite slipping of fluid on fluid across the 

 surface left by the dissolution of the infinitely thin barrier membrane ; constitut- 

 ing [§ 60 (m) above], as Helmholtz has shown, a " vortex sheet." The analytical 

 expression of the condition of equality between the tangential velocities is that 

 the variation of the velocity potential in tangential directions shall be equal on 

 the two sides of each barrier. Hence, by integration, we see that the difference 

 between the values of the velocity potential on the two sides must be the same 

 over the whole of each barrier. This condition requires that the initiating pres- 

 sure be equal over the whole membrane. For, at any time during the instituting 

 of the motion, let p v p 2 be the pressures at two points P v P 2 of the fluid, and 

 moving with the fluid, infinitely near one another on the two sides of one of 

 the membranes, so that the pressure «r, which must be applied to the membrane 

 to produce this difference of fluid pressure on the two sides, is equal to^ — p 2 in 

 the direction opposed to p v And let <p L , <p 2 be the velocity potentials at P x and 

 P 2 , so that if fds denote integration from P 1 to P 2 , along any path I^PPa what- 

 ever from P x to P 2 , altogether through the fluid (and therefore cutting none of 

 the membranes), and £ the component of fluid velocity along the tangent at any 

 point of this curve, we have 



f£ds = p 2 ~<p 1 (1). 



Hence, by (6) of § 59, 



d (<P2 ~ Pi) _ , ifa 2 - a 2 ) (2) 



jj. > — w — 2 \1\ 1i) .... (4), 



where q v q 2 denote the resultant fluid velocities at P x and P 2 . Now, the normal 

 component velocities at P x and P 2 are necessarily equal ; and therefore, if the 

 components parallel to the tangent plane of the intervening membrane are also 

 equal, we have 



?1 = ?2 



