230 VORTEX MOTION. [CHAP. VII 



where the accents attached to , 77, f are used to distinguish the 

 values of these quantities at the point (x f , y , z \ and 



r={(x- # ) 2 + (y - y f + (z- z J}*, 



as before. The integrations are to include, of course, all places at 

 which f , 77, f differ from zero. 



It remains to shew that the above values of F, G, H really 

 satisfy (4), Since djdx .r~ l = d/dx . r~ l , we have 



dF dG dH 1 ffff d , d ., d \ 1 , , , , , , 

 j- + -j- + -j- = ~ a r I ? V-&amp;gt; + V j-/ + ? j~&amp;gt; - dxdy dz 

 dx dy dz Sirj/Jv ** ^2/ dz J r 



7T~ I I (1% + m? ? / + ?l ? ) \~ cT 



ZTT J J r ZTT 



(7), 



by the usual method of partial integration. The volume-integral 

 vanishes, by Art. 142 (i), and the surface-integral also vanishes, 

 since lj~ + 77177 + nf = at the bounding surfaces of the vortices. 

 Hence the formulae (3) and (6) lead to the prescribed values of 

 f, 77, f, and give a zero velocity at infinity. 



The complete solution of our problem is now obtained by 

 superposition of the results contained in the formula (1) and (3), 



viz. we have 



d$&amp;gt; dH dG 



u = -^j + ^j j- 1 



dx dy dz 



d3&amp;gt; dF dH 



d&amp;lt;$&amp;gt; dG dF 



w = - -r + -= -j- 



dz dx dy 



where &amp;lt;l&amp;gt; ; F, G, H have the values given in (2) and (6). 



When the region occupied by the fluid is not unlimited, but is bounded (in 

 whole or in part) by surfaces at which the normal velocity is given, and when 

 further (in the case of a cyclic region) the value of the circulation in each of 

 the independent circuits of the region is prescribed, the problem may by a 

 similar analysis be reduced to one of irrotational motion, of the kind con 

 sidered in Chap, in., and there proved to be determinate. This may be left 

 to the reader, with the remark that if the vortices traverse the region, 

 beginning and ending on the boundary, it is convenient to imagine them 

 continued beyond it, or along its surface, in such a manner that they form 

 re-entrant filaments, and to make the integrals (6) refer to the complete system 

 of vortices thus obtained. On this understanding the condition (4) will still 

 be satisfied. 



