508 Prof. Helmlioltz on Integrals expressing Vortex-motion. 



As long as we consider a finite number of separate and indefi- 

 nitely thin vortex-rings in these sums, we can only introduce in 

 w, t, and yjr the portions produced by the other rings. If, how- 

 ever, we suppose the space to be continuously filled with an infi- 

 nite number of such, ^ is the potential function of a continuous 

 mass, w and r its differential coefficients ; and we know that for 

 such a function, as well as for its differential coefficients, the 

 parts due to an indefinitely small portion indefinitely near the 

 point for which the value is sought, are indefinitely small in 

 comparison with those of finite masses at a finite distance*. 



Let us change the sums into integrals : we may suppose the 

 entire amount of their value at any point to be expressed by w> 

 t, and tyj and put 



dX dp 



For this purpose we express m as adp dX, 



rp^dpdX=0, (9) 



2\\ap^dpdX-\\apX^dpdX^UcTp^dpdX. {9a) 



Since the product adp dX is by § 2 constant with respect to the 

 time, (9) can be integrated with respect to t, and we have 



l-ffap-dp dX= const. 



Consider the space to be divided by a plane which passes 

 through the axis of z, and therefore cuts all the vortex-rings ; 

 consider a as the density of a slice of the mass, and call Wl 

 the entire mass in the slice made by the plane; 



Wl=ff*dpdX; 



and if K 2 is the mean value of p 2 for all the elements of mass, 



ffcrp.pdpdX = m}tf; 



and since this integral and the value of 9tt remain unchanged 

 during the lapse of time, R also remains unchanged during the 

 motion. 



Therefore if there exist in the unlimited fluid only one circular 

 vortex-filament of indefinitely small section, its radius remains 

 unaltered. 



The magnitude of the vis viva is in our case, by (6 c), 



* Compare Gauss in the Resultate des magnctischen Vereins, 183.9, p. 7. 



