59 Gl.] THEOREMS MODIFIED BY MULTIPLICITY. 55 



The theorems of Art. 48 also call for modification. The proper 

 extension of (/3) is as follows : 



61. A function &amp;lt; exists which satisfies (10) throughout a given 

 n + 1 -ply- connected region, which has any given cyclic constants 

 K iy K z ,...K n corresponding to the n independent non-evanescible cir 

 cuits capable of being drawn in the region, and which is such that its 



rate of variation ,- in the direction of the normal has a given value 



at every point of the boundary. These arbitrary values of ~ must 



diii 



of course fulfil the condition H~dS=0. 



JJdn 



\Ve follow Thomson in marshalling the following physical consi 

 derations in support of this theorem. Let us suppose the region oc 

 cupied by incompressible fluid of unit density enclosed in a perfectly 

 smooth and flexible membrane. Further, let n barriers be drawn, 

 as in Art. 54, so as to reduce the region to a simply-connected one, 

 and let their places be occupied by similar membranes, infinitely 

 thin, and destitute of inertia. The fluid being initially at rest, let 

 each element of the first-mentioned membrane be suddenly moved 



inwards with the given (positive or negative) normal velocity -^ , 



diii 



whilst uniform impulsive pressures K^ K 2 , ... -K n are applied to 

 the positive sides of the respective barrier-membranes. Some 

 definite motion of the fluid will ensue, characterized by the follow 

 ing properties : 



(a) It is irrotational, being generated from rest ; 



(b) The normal velocity at every point of the original bound 

 ary has the assigned value ; 



(c) The values of the impulsive pressure, and therefore of the 

 velocity-potential, at two adjacent points on opposite sides of a 

 barrier-membrane, differ by the corresponding value of K, which is 

 constant over the barrier ; 



(d) The motion on one side of a barrier is continuous with 

 that on the other. 



To prove the last statement we remark, first, that the velocities 

 normal to the barrier at two adjacent points on opposite sides of it 



