SIMILAR TO THOSE OF HYDRODYNAMICS. 



371 



du 



For steady motion the terms -57 vanish, and then multiplying these equations by 

 t<! , k 2 . . . u u respectively and adding, we have 



dW . dW 



dW 

 'i dx x 



~ «a dxZ + * • • + «* ^r = 



Equations (31) may be written in the form 



dW du k 



dx k 



lit 



+ H„ 



(32) 



(33) 



Then, by taking equation (5) into account, we have for the determination of W 



k=n 



V 2 TF=2 



k-l 



dx k 



(34) 



Take now any two of equations (10), say those indicated by the subscripts j and k ; 

 differentiate the ^'-equation for x,. and the ^-equation for x J; and subtract the second 

 result from the first; continue the process until all of equations (10) have been com- 

 bined two at a time by this process ; we thus arrive at a group of — ^ — equations 

 of the form 



dtn 

 dt 



dt. ? 'l dr. ^ V i dx, ' 



dx x 



du Y du. 



(tX 4 (tx._ 



""*" "» dx n 

 du. 



du. 



+ £12 ( dx, + dx, I + &3 dx., + £u dx« + 



du n 



du-. 



£23 dx, K 



du 4 



dx x = 24 dx L 



+ 



r7« 



"=0 





or generally 



W + »i ^ + «2 ^ + ' 



4_ t dUx 1- f -"- 2 4_ 

 "t" Si* da;, "f" £2* ^ i" 



* duj j. du 2 



~~ &J dx, ~~ &W ~dx k ~ 



+ Ik 



» cfo, 



+ 6 



du n 

 ~&» dx,- 



+ 6 





2' j_ *L*\ 



•,■ "^ dx k J 



(35) 



£ — "=0 



Since ^ = £ kk = and ^ = — £ kJ , we may write this equation in the form 



~S* 1 1 



- + 2 



mmjf 



= 



(36) 



These equations give the vortex motion in the n-dimensional space ; for n = 3, 

 they obviously become the ordinary equations for vortex motion. 



