252 SIR WILLIAM THOMSON ON VORTEX MOTION. 



two such closed curves if mutually irreconcilable (§ 58) ; but different mutually 

 reconcilable closed curves will not be called different circuits. 



60. (t). Thus, (?z + l)ply continuous space, is a space for which there are n, and 

 only n, different circuits. This is merely the definition of § 58, abbreviated by 

 the definite use of the word circuit, which I now propose. The general termin- 

 ology regarding simply and multiply continuous spaces is, as I have found since 

 § 58 was written, altogether due to Helmholtz ; Riemann's suggestion, to which 

 he refers, having been confined to two-dimensional space. I have deviated some- 

 what from the form of definition originally given by Helmholtz, involving, as it 

 does, the difficult conception of a stopping barrier;* and substituted for it the 

 definition by reconcilable and irreconcilable paths. It is not easy to conceive the 

 stopping barrier of any one of the first three diagrams of § 58, or to understand 

 its singleness ; but it is easy to see that in each of those three cases, any two 

 closed curves drawn round the solid wire represented in the diagrams are recon- 

 cilable, according to the definition of this term given in § 58, and therefore, that 

 the presence of any such solid adds only one to the degree of continuity of the 

 space in which it is placed. 



(u). If we call a partition, a surface which separates a closed space into two 

 parts, and, as hitherto, a barrier, any surface edged by the boundary of the space, 

 Helmholtz's definition of multiple continuity may be stated shortly thus : — 



A space is (n + l)p!y continuous if n barriers can be drawn across it, none of 

 which is a partition. 



(v). Helmholtz has pointed out the importance in hydrokinetics of many- 



—i y 

 valued functions, such as tan -^, which have no place in the theories of gravi- 

 tation, electricity, or magnetism, but are required to express electro-magnetic 

 potentials, and the velocity potentials for the part of the fluid which moves irro- 

 tationally in vortex motion. It is, therefore, convenient, before going farther, 

 that we should fix upon a terminology, with reference to functions of that kind, 

 which may save us circumlocutions hereafter. 



(w). A function <f> (%, y, z) will be called cyclic if it experiences a constant 

 augmentation every time a point P, of which x, y, z are rectangular rectilineal 

 co-ordinates, is carried from any position round a certain circuit to the same 



position again, without passing through any position for which either -~, -~, or 

 -p becomes infinite. The value of this augmentation will be called the cyclic 



* But without this conception we can make no use of the theory of multiple continuity in 

 hydrokinetics (see §§ 61-63), and Helmholtz's definition is, therefore, perhaps preferable after all 

 to that which I have substituted for it. Mr Clerk Maxwell tells me that J. B. Listing has more 

 recently treated the subject of multiple continuity in a very complete manner in an article entitled 

 " Der Census raumlicher Complexe." — Konigl. Ges. Gottingen, 1861. See also Prof. Cayley "On 

 the Partition of a Close." — Phil. Mag. 1861. 



