144-145] KINEMATICAL THEORY. 229 



r denoting the distance between the point (x f , 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 lt w l are required, viz. 



r = {(x - xj + (y- yj +(z- /)}*, 



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

 from zero. 



To verify this result, we notice that the above values of u l} v lt w l 

 make 



<^ + j, 1+ <^ = _ v 



dx ay dz 

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



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



we assume 



dH dG dF dH dG dF /QN 



U = J 7-, V 2 = -T 7, W 2 =-J If | (O), 



2 dy dz dz dx' dx dy 



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

 conditions. In the first place, these formulae make 

 du 2 dv 2 dw 2 _ n 



~~T 1 if r ~J ") 



dx dy dz 



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

 give 



2 = ^ 2 _ 2 = ( d + + 1 - V*F 

 * dy dz dx \dx dy dz ) 



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

 F, G } H satisfying 



dF + dG + dH_ =Q 4) 



dx dy dz 



and V 2 .P=-2f, V*G = -2rj, V*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, 7//27T, f/27r, respectively ; thus 



.(6), 



