488 Prof. Helmholtz on Integrals expressing Vortex-motion* 



Hitherto, with scarcely an exception, no cases have been treated 

 except those where not only have the forces X, Y, Z a potential 

 Y, so that 



dV Y dV dV 



ax ay dz 



but also a velocity-potential </> can be found, giving 



dd> d<b d<f> 



u-= -~i v= -f-j W=-~-' . . . . (1 0) 

 dx dy dz v ' 



By these assumptions the problem is immensely simplified, 

 since the first three of the equations (1) give a common integral 

 from which p is to be found, (/> having previously been determined 

 so as to satisfy the fourth equation, which becomes in this case 



dx* ■ dtf '"*" dz* ' 



and coincides with the known differential equation for the po- 

 tential of magnetic masses which are external to the space in 

 which the equation holds good. And it is known that every 

 such function <£ which satisfies the above equation within a sim- 

 ply-connected* space, can be expressed as the potential of a defi- 

 nite distribution of magnetic matter on the bounding surface, as 

 I have already mentioned in the introduction. 



In order that the substitution (1 b) may be lawful, we must 

 have 



du dv _ dv ^dw ___ dw du _ 



dy dx~ ' dz dy~ y dx dz~ ' " V ' 



To understand the mechanical meaning of these last three 

 conditions, we may consider the changes undergone by an inde- 

 finitely small volume of the fluid during the time dt as com- 

 pounded of three separate motions. 1st. A translation of the 

 whole in space. 2nd. An expansion or contraction of the whole 

 parallel to three dilatation-axes, so that any rectangular paral- 

 lelepiped whose edges are parallel to these axes may remain 

 rectangular, while its edges alter their length but remain pa- 

 rallel to their original directions. 3rd. A rotation about some 

 instantaneous axis, which, as we know, may be considered as 

 the resultant of three rotations about the axes of coordinates. 



If the conditions (1 c) are fulfilled at a point whose coordi- 



* In complexly-connected spaces ^> may have more values than one ; 

 and for multiple-valued functions which satisfy the above differential equa- 

 tion Green's fundamental theorem does not hold ; and hence a great num- 

 ber of its consequences which Gauss and Green have deduced for magnetic 

 potential functions also fail, since the latter, from their very nature, can have 

 but single values. 



