GENERAL PROPERTIES OF CERTAIN PARTIAL DIFFERENTIAL EQUATIONS, ETC. 363 



intention to generalize someAvhat the methods and results of Part I. to make an 

 application to the general theory of steady motion (n variables), and finally to apply 

 all of the results of both parts to space of three dimensions, i. e. to the ordinary 

 theory of hydrodynamics. 



(Just here I wish to call attention to the Note at the end of this paper; it 

 properly belongs in Part II., but as I wish it made public at the earliest opportunity, 

 1 have inserted it in Part I.) 



In Part I. an account is given of a set of partial differential equations similar to 

 those of hydrodynamics, but containing n + 1 independent variables, or, if we may 

 use the expression, the equations of fluid motion for space of n dimensions. 



Denote by Xj, x 2 , x s , . . . . x n , t the independent variables of the problem; by 

 u i} ?< 25 • • • • m»j U, certain functions of these variables. For a problem in hydrody- 

 namics the function U is given by 



U 



= r+f*. 0) 



V being the potential of external forces. The functions u are similar to the velocities 

 in such an ordinary problem. Write 



2q = u? + ui + .... + u n \ (2) 



then 



tin till. rJ 11 ,1*. 



(3) 



(4) 



of these quantities. The functions u are supposed to be connected by the relation 



du, du, 4- d "»-() (RS 



<&, + dx 2 + •••• + Jx n - °- ( 5 ) 



Denoting now the operator 



dxf T ,%,;% -T- f dx * 



by y, we have, on observing that g Jk = — g v , 



