191, 192] VORTEX -MOTION. 511 



Now at an instant later by dt, when the particles are at A' and B f , 

 we have 



dx' = dx + (U B - O dt = 8 + 



63) dy' = dy + (u* - U A } dt = sQ+ g) dt\ 



de' = dz + (U B - U A ) dt = e[ + g ) dt] 



Therefore the projections of the arc ds' in the new position are 

 proportional to the new values of > ? > as they originally were, 



so that the particles still lie on a vortex -line. Accordingly a vortex- 

 line is always composed of the same particles of fluid. Also since 

 the components of ds have increased or have changed so as to be 



always proportional to the components of ? if the liquid is in- 



compressible the rotation is proportional to the distance between the 

 particles. And whether Q vary or not, if S be the area of the cross- 

 section of a vortex -filament, gSds, the mass of a length ds remaining 

 constant, so does So, the strength of the filament. 



It is easy to see that this is equivalent to a statement of the 

 conservation of angular momentum for each portion of the fluid. 

 Evidently in a perfect fluid no moment can be exerted on any portion, 

 since the tangential forces vanish. 



Accordingly the strength of a vortex -filament is constant, not 

 only at all points in the filament but at all times, consequently a 

 vortex existing in a perfect fluid is indestructible, however it may 

 move. It is from this remarkable property of vortices discovered 

 by Helmholtz that Lord Kelvin was lead to imagine atoms as COD- 

 sisting of vortices in a perfect fluid. 



192. Vector Potential. Helmholtz's Theorem. We have 

 seen that any curl is a solenoidal vector. We may naturally ask 

 whether conversely any solenoidal vector can be replaced by the curl 

 of another vector. It was shown by Helmholtz that any uniform 

 continuous vector point -function vanishing at infinity can be expressed 

 as the sum of a lamellar and a solenoidal part, and the solenoidal 

 part may be expressed as the curl of a vector point-function. A 

 vector point -function is completely determined if its divergence and 

 curl are everywhere given. Let q be the given vector, which in our 

 case is the velocity of the fluid. Let us suppose that it is possible 

 to express it as the sum of the vector -parameter of a scalar func- 



