SIR WILLIAM THOMSON ON VORTEX MOTION. 243 



58. Adopting the terminology of Riemann, as known to me through Helm- 

 holtz, I shall call a finite position of space ft-ply continuous when its bounding 

 surface is such that there are n irreconcilable paths between an ytwo points in 

 it. To prevent any misunderstanding, I add (1), that by a portion of space I mean 

 such a portion that any point of it may be travelled to from any other point of 

 it, without cutting the bounding 'surface ; (2), that the " paths" spoken of all lie 

 within the portion of space referred to ; and (3), that by irreconcilable paths 

 between two points P and Q ; I mean paths such, that a line drawn first along 

 one of them cannot be gradually changed till it coincides with the other, being 

 always kept passing through P and Q, and always wholly within the portion of 

 space considered. Thus, when all the paths between any two points are recon- 

 cilable, the space is simply continuous. When there are just two sets of paths, 

 so that each of one set is irreconcilable with any one of the other set, the space 

 is doubly continuous ; when there are three such sets it is triply continuous, and 

 so on. To avoid circumlocutions, we shall suppose S to be the boundary of a 

 hollow space in the interior of a solid mass, so thick that no operations which we 

 shall consider shall ever make an opening to the space outside it. A tunnel through 

 this solid opening at each end into the interior space constitutes the whole space 

 doubly continuous ; and if more tunnels be made, every new one adds one to the 

 degree of multiple continuity. When one such tunnel has been made, the surface 

 of the tunnel is continuous with the whole bounding surface of the space con- 

 sidered ; and in reckoning degrees of continuity, it is of no consequence whether 

 the ends of any fresh tunnel be in one part or another of this whole surface. 

 Thus, if two tunnels be made side by side, a hole anywhere opening from one of 

 them into the other adds one to the degree of multiple continuity. Any solid 

 detached from the outer bounding solid, and left, whether fixed or movable in the 

 interior space, adds to the bounding surface an isolated portion, but does not in- 

 terfere with the reckoning of multiple continuity. Thus, if we begin with a simply 

 continuous space bounded outside by the inner surface of the supposed exter- 

 nal solid, and internally by the boundary of the detached solid in its interior, 

 and if we drill a hole in this solid we produce double continuity. Two holes, 

 or two solids in the interior each with one hole (such as two ordinary solid 

 rings), constitute triple continuity, and so on. A sponge-like solid whose 

 pores communicate with one another, illustrates a high degree of multiple con- 

 tinuity, and it is of no consequence whether it is attached to the external 

 bounding solid or is an isolated solid in the interior. Another type of multiple 

 continuity, that presented by two rings linked in one another, was referred 

 to in § 57. 



When many rings are linked into one another in various combinations, there 

 are complicated mutual intersections of the several partial barriers /3 X , /3 2 , . . „ 

 required to stop all multiple continuity. But without having any portion of the 



