55 57.] CYCLIC CONSTANTS. 51 



5G. Let $ be the flow from a fixed point A to a variable point 



P, viz. 



//&amp;gt; 



(f&amp;gt; = I (udx + vdy + wdz] (15). 



So long as the path of integration from A to P is not specified, &amp;lt;f&amp;gt; 

 is indeterminate to the extent of a quantity of the form (14). 



If however n barriers be drawn in the manner explained in 

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

 and if the path of integration in (15) be restricted to lie within 

 the region as thus modified (i.e. it is not to cross any of the 

 barriers), then &amp;lt;j) becomes a single-valued function, as. in Art. 42. 

 It is continuous throughout the modified region, but its values 

 at two adjacent points on opposite sides of a barrier differ by 

 + K. To derive the value of (/&amp;gt; when the integration is taken along 

 any path in the unmodified region we must add the quantity 

 (14), where any p denotes the number of times this path crosses 

 the corresponding barrier. A crossing in the positive direction of 

 the circuits interrupted by the barrier is here counted as positive, 

 a crossing in the opposite direction as negative. 



By displacing P through an infinitely short space parallel to 

 each of the co-ordinate axes in succession, we find 



d&amp;lt;b d(fr d6 



u=~, = W = ~Y-\ 

 dx dij dz 



so that &amp;lt; satisfies the definition of a velocity-potential, Art. 22. 

 It is now however a many-valued or cyclic function; i.e. it is 

 not .possible to assign to every point of the original region a de 

 finite value of &amp;lt;/&amp;gt;, such values forming a continuous system. On 

 the contrary, whenever P describes in the region a non-evanescible 

 circuit, &amp;lt;f&amp;gt; will not, in general, return to its original value, but 

 will differ from it by a quantity of the form (14). The quantities 

 iCj, K 2 j...K n are called by Thomson the cyclic constants of &amp;lt;. 



57. The foregoing theory is illustrated by Ex. 4, Art. 35. 

 The formulae there given make the velocity infinite at points on 

 the axis of z t which must therefore be excluded from the region to 

 which our theorems apply. This region becomes thereby doubly- 

 connected, for we can connect any two points A, B of it by two 



42 



