242 SIR WILLIAM THOMSON ON VORTEX MOTION. 



gradual variations of one curve between P and Q, each lying wholly within S. 

 Now, in a simply continuous space, a curve joining the points P and Q may be 

 gradually varied from any curve PCQ to any other PC'Q, and therefore if the 

 space contained within S be simply continuous, the difficulty depending on the 

 multiplicity of value of <p or <p' cannot exist. And however multiply continuous 

 (§ 58) the space may be, the difficulty may be evaded if we annex to S a 

 surface or surfaces stopping every aperture or passage on the openness of which 

 its multiple continuity depends ; for these annexed surfaces, as each of them 

 occupies no space, do not disturb the triple integrations (1), and will, therefore, 

 not alter the values of its first member ; but by removing the multiplicity of con- 

 tinuity, they free each of the integrations by parts, by which its second or third 

 members are obtained, from all ambiguity. To avoid circumlocution, we shall 

 call j8 the addition thus made to S ; and further, when the space within S is 

 (§ 58) not merely doubly but triply, or quadruply, or more multiply, continuous, 

 we shall designate by /3 X , /3 2 ; or f$ 1 , (3 2 , j3 3 ; and so on ; the several parts of /3 re- 

 quired in any case to stop all multiple continuity of the space. These parts of /3 

 may be quite detached from one another, as when the multiple continuity is that 

 due to detached rings, or separate single tunnels in a solid. But one part /3 a may 

 cut through part of another, /3 2 , as when two rings (§ 58, diagram) linked into one 

 another without touching constitute part of the boundary of the space considered. 

 And we shall denote byj/ds, integration over the surface /3, or over any one of 

 its parts, ft 1 , /3 2 , &c. Let now P and Q be each infinitely near a point B, of (3, but 

 on the two sides of this surface. Let k denote the value of /Yds along any curve 

 lying wholly in the space bounded by S, and joining PQ without cutting the 

 barrier; this value being the same for all such curves, and for all positions of B 

 to which it may be brought without leaving /3, and without making either P or Q 

 pass through any part of ft. That is to say, k is a single constant when the space 

 is not more than doubly continuous ; but it denotes one or other of n constants 

 k v f 2 , . . . k m which maybe all different from one another, when the space is ft-ply 

 continuous. Lastly, let k denote the same element, relatively to <p\ as k relatively 

 to <p. We find that the first steps of the integrations by parts now introduce, 

 without ambiguity, the additions 



^xf/ds &p' » and Ixffdc, ijf> (6), 



to the second and third numbers of (1) : 2 denoting summation of the integra- 

 tions for the different constituents /3 X , /3 2 , . . . of (3; but only a single term when 

 the space is (§ 58) not more than doubly continuous. Green's theorem thus 

 corrected becomes 



Ifi%Tx + fy% + Tzt) dxdydz =/f d 'W + vJJ'W -Jff^f'dxdydz 



— 1 1 da p'tjcp + 2x' // d(ts<p — II I <p v' 1 <p dx dy dz . (7). 



