5759.] CYCLIC VELOCITY-POTENTIAL. 53 



without destroying the continuity of the region. The number of 

 these cannot, by definition, be greater than n. Every other equi- 

 potential surface which is not closed will be reconcileable (in an 

 obvious sense) with one or more of these barriers. A curve drawn 

 from one side of one of these barriers round to the other, without 

 meeting any of the remaining barriers, will cross every surface 

 recoucileable with it an odd number of times, and every other sur 

 face an even number of times. Hence the circulation in the cir 

 cuit thus formed will not vanish, and will be a cyclic function. 



In the method adopted above we have based the whole theoiy 

 on the equations 



^_^_n ^_^-o dv du -(\ 

 dy dz~&quot; dz dx~&quot;&amp;gt; dx dy~ 



and have deduced the existence and properties of the velocity- 

 potential in the various cases as necessary consequences of these. 

 In fact, Arts. 41, 42, and 53 56, may be regarded as a treatise on 

 the integration of this system of differential equations. 



The integration of (16), when we have, on the right-hand side, 

 instead of zero known functions of x y y, z, will be treated in 

 Chapter vi. 



59. If the density of the fluid be either constant or a function 

 of the pressure only, and if the external impressed forces have a 

 single-valued potential, the cyclic constants of (j&amp;gt; do not alter with 

 the time. For if &amp;lt; be the initial value of the velocity-potential, 

 we have, Art. 23, 



= 00 + %&amp;gt; 



where, Art. 19 (25), 



Under the circumstances stated % is a single-valued function, and 

 the cyclic constants of (f&amp;gt; are the same as those of $ . In other 

 words the circulations in the several circuits of the region occupied 

 by the fluid are constant. 



