PAPER BY PROF. HELMHOLTZ. 55 



Therefore if we change the sums into integrals we can understand 

 by iv, T, and //: the total values of these quantities that exist at the 

 point in question, and can put 



d\ do 



dt ' dt 



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

 the summations thus become converted into the following integrals : 



//ffp'l^dpdX = (9) 



Since, in accordance with Sect, ii, the product ff dp dX does not vary 

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

 to t, and we obtain 



hJ^J^ap^ dp d\ — Const. 

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

 z, aud therefore intersects all the vortex rings that are present j then 

 consider a as the density of one layer of the mass, and let i1f be the 

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



m=ffGdpdX, 



and let E^ be the mean value of p~ for all the elementary masses, then 



fj V/Q. pdp dX = m R-, 



and, since this integral and the value of ^)l do not vary with the time, 

 it follows that R also remains unchanged during the motiou 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 y(k), the total living force in our case is 



K=-hff/{La+Mr])da dh dc. 



= —h/'/'/'ipa'p dp dX de. 



= —InldrCipG-p dp dX, 



This also does not change with time. 



Furthermore, because o dpdX does not vary with time, therefore, 



^Jfop'X dp dX = IffapX ^f dpdX^ffap^ § dX dp ; 



therefore if we indicate by / the value of A for the center of gravity of 

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

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

 given, we obtain 



'^dtf f^P"^^P'^^-^~^f f''P^^-^')Tt^i''^^^-^h • ' (^^^ 



