372 



GENERAL PROPERTIES OF CERTAIN PARTIAL DIFFERENTIAL EQUATIONS 



The equations giving the motion of a fluid particle are, of course, 



da\ dx, 



dt 



ui,- d j = Us 



dx n 



• dt 



«,. 



(37) 



Writing for brevity 

 equations (30) become 



D _ d d d d 



dt. — d+~~ u i Tfc. ~> "a jlZ"'" u " dx a 



dt dt 



dU Dii! 



dx x 



dx, 



dU _ Dih 



dx — dt > etC> 



</./-, " dt 

 Multiplying these by dx x , dx 2 . . . and adding, we have readily 



dU -\- dq = -j t (u^Ixy + u.,dx 2 + • • ■ u n dx„). 



Integrating from a point a to another yS, we have 



diJa (ihdXi + itodx., + • • • v u dx„) = [U+ q]l. 



If a coincide with fi, i. e. if we integrate around a closed curve, 



— j\Uydx x + u 2 dx 2 + • • • + u n dx n ) — 0, 



(38) 



(39) 



(40) 



(41) 



a theorem corresponding to that in hydrodynamics which states that the circulation in 

 any closed curve moving with the fluid is independent of the time. As the principal 

 object of this paper is to carry on and generalize the investigations begun in the 

 paper on steady motion previously referred to, I shall leave for the present any 

 further examination of the properties of the quantities dealt with so far and begin 

 from a rather more general point of view. In steady motion a certain Jacobian ap- 

 peared which entered into all of the work in a curious manner, this Jacobian was 

 found from the functions lf 2 , 3 , i. e. 



= J. 



In the present more general problem we may introduce a set of functions 



6 ,01,02 ... . n -\ 



(42) 



