PAPER BY PROF. HELMHOLTZ. 55 



Therefore if we change the sums into integrals we can understand 

 by to, r, and ip the total values of these quantities that exist at the 

 point in question, and can put 



dX dp 



dt ' dt 



To this end we replace the quantity m by the product adpdX, and 

 the summations thus become couverted into the following integrals : 



f/<jp%dpdX = (9) 



2ff*P t §dpd\-ff<jpxigapdX=ff G pl,dpdX . . (9a) 



Since, in accordance with Sect, n, the product adpdXdoes not vary 

 with the time, therefore, the equation (9) can be integrated with respect 

 to t, and we obtain 



\J' \ftip 2 dp dX = Const. 

 Imagine the space divided by a plane that passes through the axis of 

 z, and therefore intersects all the vortex rings that are present; then 

 consider a as the density of one layer of the mass, and let 9Ji be the 

 total mass in this layer adjoining this dividing plane ; therefore, 



W.=fj 1> GdpdX, 



and let B 2 be the mean value of p 2 for all the elementary masses, then 



/J'ffp.pdpdX=MB 2 , 



and, since this integral and the value of 2Ji do not vary with the time, 

 it follows that B also remains unchanged during the motion of transla- 

 tion. 



Therefore if there exists in the unlimited mass of liquid only one 

 circular vortex filament of infinitely small sectional area, then its radius 

 remains unchanged. 



According to equation (6c), the total living force in our case is 



K=-lifff{L$+Mrj)da db dc. 



=i — Urj , J'q6 , p dp d_X de. 



= —InlifJ'^o-pdpdX. 



This also does not change with time. 



Furthermore, because a dp dX does not vary with time, therefore, 



%ffaf?X dp dX = 2/fapX % dp dX+ffap 2 1 dX dp ; 



therefore if we indicate by I the value of X for the center of gravity of 

 the vortex filament treated of in equation (9a), and multiply (9) by this 

 I, and add the result to (9a), and substitute therein the equation last 

 given, we obtain 



7 T\ 



^Lf/^ 2Xd P <1X+r °ff 0f){l - X) Tt dpdX = '^h ' ' m 



