24 Mr M. J. M. Hill, On some General Equations. [Oct. 29, 

 it is shewn that if n be odd 



dx t dx 2 ' ' dx n 

 and n equations, of which the following is a type, 



fd d_ 



\dt 1 dx 1 



'd . d , d J , d \ /£. 



p cfo? r p c&£ 2 p dx n ' 



When ft is even, these reduce to identities and are replaced by 

 the single equation 



id d d d \ /JB\ n 



3. Adopting the language of Fluid Motion it is shewn that 

 the vortex lines 



ax. ax„ ax n 



tl ?2 »re 



always contain the same particles. 



4. It is shewn that u l dx 1 + ... + u n dx n (if n be odd) may always 

 be reduced to the form 



dK+P n _ ±1 dP 1 + P n ^dP, + ...+P n _ 1 dP 



22 2 



which is Clebsch's form (see his paper " Ueber die Integration der 

 hydrodynamischen Gleichungen." Crelle, Bd. lvi.) ; and the mean- 

 ing of this form is that the vortex lines are the intersections of the 

 (n — 1) loci 



P lt P 2 , . . . P„_ t , P n+1 . . . P n _ v = constant, respectively. 



