Prof. Helmholtz on Integrals expressing Vortex-motion. 509 



K = - hfff{Jj% + M77) da db dc 



= — hfffyjrapdp dX de 



= —■ZirhfJ'^rcrpdp dX. 



This is also constant as regards time. 



Again, we remark that, as <rdp dX is constant with respect to t 3 



j t j (ap' 2 XdpdX = 2 J Y&pk&dp d\+ Uap^ — dXdp, 



hence the equation (9 a), if we call I the value of X for the cen- 

 tre of gravity of the section of the vortex-filament and multiply 

 (9) by it and add, becomes 



2 iJjVxrfp d\ + 5^ap(l-X) g dpdX=- ~. (9 b) 



If the section of the vortex-filament is indefinitely small and e an 

 indefinitely small magnitude of the same order as l—X and the 

 other linear dimensions of the section, but adp dX finite, then yfr 

 and K are of the same order of indefinitely great quantities as 

 log e. For very small distances v from the vortex- ring we have 



v 2 

 V 



In the value of K, yjr is multiplied by p or g. If g is finite and 

 v of the same order as e, K is of the order log e. Only when g 



is indefinitely great and of the order - does K become indefi- 

 nitely great, of the order - log e. Then the circle becomes a 

 straight line. But, on the other hand, ~. which is equal to -j-, 



1 



dV * dz 



becomes of the order -, the second integral therefore is finite, 



and for finite values of p is indefinitely small compared with K. 

 In this case we may put in the first integral / instead of X, and 

 we find 



27rh 

 Since 90? and R are constants, / must vary proportionally to the 

 time. If Wl is positive, the motion of the elements of fluid on 



