151-152] KINETIC ENERGY. 241 



o-Ss and o-'Ss', respectively, so that the expression following the 

 integral signs in (2) is equivalent to 



cos e ^ / /ex / 



- . Q)(TOS . (i) (7 OS . 



r 



where e is the angle between &s and 8s'. If we put &>o- = m, 

 wo-' ' = m' ', so that m and m' denote the strengths of the two 

 elementary vortices, we have 



(3), 



where the double integral is to be taken along the axes of the 

 filaments, and the summation embraces every pair of such 

 filaments which are present. 



The factor of p/w in (3) is identical with the expression for the 

 energy of a system of electric currents flowing along conductors 

 coincident in position with the vortex-filaments, with strengths 

 m, m',... respectively*. The above investigation is in fact merely 

 an inversion of the argument given in treatises on Electro- 

 magnetism, whereby it is proved that 



dsds > = jLf/7 '( a * + p + y.) dxdydz, 



i, i' denoting the strengths of the currents in the linear conductors 

 whose elements are denoted by &s, Ss', and a, /3, 7 the components 

 of magnetic force at any point of the field. 



The theorem of this Art. is purely kinematical, and rests solely 

 on the assumption that the functions u, v, w satisfy the equation 

 of continuity, 



du dv dw_ ft 



dx dy dz 



throughout infinite space, and vanish at infinity. It can therefore 

 by an easy generalization be extended to the case considered in 

 Art. 141, where a liquid is supposed to circulate irrotationally 

 through apertures in fixed solids, the values of u, v, w being now 

 taken to be zero at all points of space not occupied by the fluid. 

 The investigation of Art. 149 shews that the distribution of velocity 

 thus obtained may be regarded as due to a system of vortex-sheets 

 coincident with the surfaces of the solids. The energy of this 

 system will be given by an obvious adaptation of the formula (3) 

 above, and will therefore be proportional to that of the correspond- 



* See Maxwell, Electricity and Magnetism, Arts. 524, 637. 

 L. 16 



