144-145] KINEMATICAL THEORY. 229 



r denoting the distance between the point (# , y , z) at which the 

 volume-element of the integral is situate and the point (x, y, z) 

 at which the values of u l ,v l , w^ are required, viz. 



r = a -* 



and the integration including all parts of space at which 6 differs 

 from zero. 



To verify this result, we notice that the above values ofu lt v lf w l 

 make 



^ + ^ 1+ M = _ V2$= 



dx ay dz 

 by the theory of Attractions, and also vanish at infinity. 



To find the velocities (u. 2 , v. 2 , w 2) say) due to the vortices, 

 we assume 



_dll dG _dF dH _dG dF 



U 2 ~~d^~dz V 2 ~~dz~ dx&amp;gt; W *~dx~dy&quot;&quot; 



and seek to determine F, G, H so as to satisfy the required 

 conditions. In the first place, these formula) make 



du 2 d/Vz dw 2 _ Q 

 da dy dz 



and so do not interfere with the result contained in (1). Also, they 

 give 



2 = ^ 2 _ 2 = . 



dy dz dx\dx dy dz 



Hence our problem will be solved if we can find three functions 

 F, G, H satisfying 



dFdG^dU ..................... 



dx dy dz 



and V 2 ^=-2f, V&quot;G = -2r), V 2 H = -2 ......... (5). 



These latter equations are satisfied by making F, G, H equal to the 

 potentials of distributions of matter whose volume-densities at the 

 point (x } y, z) are f/2?r, 77/2-77-, f/2-Tr, respectively ; thus 



(6), 



