49-51] VELOCITY-POTENTIALS IN CYCLIC REGIONS. 5*7 



By displacing P through an infinitely short space parallel to 

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



d6 dd&amp;gt; d(j) 



u = ^ i r ) = y- , w = --f- ; 

 dx dy dz 



so that &amp;lt;j&amp;gt; satisfies the definition of a velocity-potential (Art. 18). 

 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 unique 

 and definite value of $, such values forming a continuous system. 

 On the contrary, whenever P describes an irreducible circuit, &amp;lt;j&amp;gt; 

 will not, in general, return to its original value, but will differ from 

 it by a quantity of the form (1). The quantities K lt /c 2 ,...fc n , which 

 specify the amounts by which &amp;lt; decreases as P describes the several 

 independent circuits of the region, may be called the cyclic con 

 stants of &amp;lt;/&amp;gt;. 



It is an immediate consequence of the circulation-theorem of 

 Art. 34 that under the conditions there presupposed the cyclic 

 constants do not alter with the time. The necessity for these 

 conditions is exemplified in the problem of Art. 30, where the 

 potential of the extraneous forces is itself a cyclic function. 



The foregoing theory may be illustrated by the case of Art. 28 (2), where 

 the region (as limited by the exclusion of the origin, where the formula 

 would give an infinite velocity) is doubly-connected ; since we can connect 

 any two points A, B of it by two irre- 

 concileable paths passing on opposite 

 sides of the axis of z, e.g. ACB, ADB in 

 the figure. The portion of the plane zx 

 for which x is positive may be taken as 

 a barrier, and the region is thus made 

 simply-connected. The circulation in 

 any circuit meeting this barrier once 



only, e.g. in ACBDA, is j^ p/r. rd0, or 2?r/i. That in any circuit not meeting 

 the barrier is zero. In the modified region may be put equal to a single- 

 valued function, viz. pd t but its value on the positive side of the barrier is 

 zero, that at an adjacent point on the negative side is -2w/i. 



More complex illustrations of irrotational motion in multiply-connected 

 spaces will present themselves in the next chapter. 



51. Before proceeding further we may briefly indicate a some 

 what different method of presenting the above theory. 



Starting from the existence of a velocity-potential as the characteristic of 

 the class of motions which we propose to study, and adopting the second 



