Prof. A. Gray: Notes on Hydrodynamics. 9 



of (18) are more compact and more general, since they are 

 applicable to any system of three axes. Moreover, they can 

 be written down at once from (14) which it is easy to 

 remember. 



11. From (18) we might deduce v. Helmholtz^s vortex- 

 motion equations, but they are perhaps most easily obtained 

 in the manner indicated below. They are introduced here 

 as the discussion will be found to lead to what seems a new 

 proof of the constancy of the moment (product of angular 

 velocity by cross-section) of a vortex-filament. 



By (1) above the equations of motion for the axes Oy, 0~ 

 are 



Bv ~dv _ BV 1 Bp 1 



'dt y~ds ~ dy p~dy 



~div , "div BV 1 "dp 



i 

 I 



(22) 



These also hold whether the axes are at right angles or not. 

 Taking the axes at right angles differentiate the first equa- 

 tion with respect to z, and the second with respect to ?/, and 

 subtract the first result from the second. We obtain easily 



b B^' oy B~ 



dt 



(23) 



here © = 



3" 3^ 



Be 



the " expansion" (time-rate of dilatation per unit of volume). 

 Two similar equations hold for 77, f and can be written down 

 at once. 



It may be remarked that the right-hand side of (23) can 

 be written as in the alternative equation 



at a *B# ox dJ 

 Similar equations hold of course for the other axes. 

 From the equation of continuity 



d -t f +pS=0 



at r 



we obtain by substitution for @ in (23) and (23') 



(23') 



- ( S )= 

 dt \pj 



?B" vbu ?3" 

 PO<<' pou poz 



I B" vo>- Zoic 

 p 0^ p 3^' p 3 



; 



(24) 



