SIMILAR TO THOSE OF HYDRODYNAMICS. 37<J 



in which 



*=?J 'fif dA 1 * dA* 



The modifications undergone by (69) and (71) in the case of steady motion are 

 obvious and need no remark. 

 The equations 



have been replaced by 

 If we have 



dx-i _ dr., 



~di — u i> ~di~ = l( 2> etc - 



dxi _ (7x. 2 _ 



~dj — An ~ai ~ A u-, etc. (/3 ; 



dt u 



then 6 k = const, is an integral of these equations, and similarly for any other of the 



functions 6. If 



X,= 

 then 



d&,. , d$ k , , rZ$,. , , d$> k , d$ k ,. 



-di = A^ + A^ + "-- + d ^ n +^f = ^ = o 



and in this case 



<£> k = const. 



is also an integral of (73). If all of the functions 6j are independent of t, we shall 



have 



Q x = const., 2 = const n _j = const. 



as integrals of (73). Suppose in equations (10) that the functions u were 



u ^/ 1 u 2 . . . . v 2u 



then it is known (see article of Professor Tanner's above referred to) that 



u dx + iiidxi +••••+ u 2n dx 2n 



can always be reduced to the form 



dM + L x dM x + L,dM 2 -\ h L n dM n 



and 



dM >nr <ni n 



«* = </,-.. + A iy k +•••• + L n -^ ( ,4) 



In this case we should also have 



d{U~g)= 2 w ^ + * 2 2 (S - g) («*** - * *3,) ( T5) 



4=0 j=0 fc=0 \ J / 



