238 VORTEX MOTION. [CHAP. VII 



velocity being everywhere finite and continuous, the parts of the 

 volume-integral on the right-hand side of (4) which are due to the 

 substance of the vortex itself may be neglected in comparison 

 with those due to the remainder of the space included within S. 

 Hence we may write 



fjjudxdydz = - f |T^ dxdydz = jjtydS + '2m ffldS' . . . (5), 



where <j> has the value given b}' Art. 147 (4), m denoting the 

 strength of the vortex (so that 2m' is the cyclic constant of <), and 

 SS' an element of any surface bounded by it. Substituting in 

 (4), we infer that the components of the impulse parallel to the 

 coordinate axes are 



2m' pf/nutS*, Zm'pffndS' ............... (6). 



Again, taking moments about Ox, 

 H$p(yZ'-zY') dxdydz 



~ zv ) dxdydz pff(ny mz) (f>dS ......... (7). 



For the same reason as before, we may substitute, for the volume- 

 integral on the right-hand side, 



= ff(ny - me) <f>dS + 2mff(ny - mz) dS' ......... (8). 



Hence, and by symmetry, we find, for the moments of the impulse 

 about the coordinate axes, 



2m'pff(ny-mz)dS', 2m'pff(lz-na;)d&, 2m' pff(mx - ly) dS' ...(9). 



The surface-integrals contained in (6) and (9) may be replaced 

 by line-integrals taken along the vortex. In the case of (6) it is 

 obvious that the coefficients of mp are double the projections on 

 the coordinate axes of any area bounded by the vortex, so that the 

 components in question take the forms 



,. ..(10). 



ds J 



