SIMILAR TO THOSE OF HYDRODYNAMICS. 383 



A u , to erase the first of equations (85) and solve the remaining n — 1 equations for 

 u 2 ,u s , . . . ., u n . We have after doing this 



\rde de.r\ u , r<w a den , u , .pw.. *».1 m ) 



t [_ * * ^. J u L * rfx J ' * ' : ■ L* rfa; J ,J i 



(89) 



Al»3 



Differentiate these with respect to x 2 , x 3 , . . . . x n and add and we readily find 



and in like manner 



VK_ A du, du, du, 



dt ~ ^u d Xl ^ A ™ dx 2 ^ ■ ■ ■ • ^ A dx n 



DA X „ _ diCj . du., _i a du > 



~dT — ^u &; + Ai2 d7 2 + — + A i* dx~ n 

 «a "^a/^^t • • • • -r Am dj . n ^ 90 ^ 



-ZH,, _ A du n dfu^ du^ 



dt — A n dx, " i " ^ 12 dx 2 "*" "*" ^ lre efe. 



If in these equations we substitute for il u the other principal minors Jj ( , igj, . . . , -4„ 4 



(i having all values from i = 1 to i = u in all of these) the equations will still hold. 



I do not write the proof of this out in full, as it is extremely simple and yet rather 



tedious. 



If we combine the equations = const, in groups of n — 1 each, we shall of 



course obtain the n lines representing the intersections of n hypersurfaces represented 



by the equations = const. Call these lines s l5 s 2 , . . . . s, t ; and denote by ds lt 



ds 2 , . . . . ds n the n sides of an element of fluid occupying the space between the 



hypersurfaces 



0i — a i > 2 = a% . . . . n = a tl 



and 



d 1 = ai -\- da if 2 = a 2 ~h da^ . . . . a — a^ + da^* 



We must have now 



o x = 1 (x 1 ,x 2 . , . .x n ,t) 



(91) 



^ + da t = 0! {x x + \ n ds! -f- Xa + Ai2^ s 2 ...•«» + Xi„^« , 



the quantities X n , X 12 , . . . . \ ln denoting the 'direction cosines' of ds x . Similar equa- 

 tions of course hold for a 2 , a s , . . . . a„. By Taylor's theorem we now obtain 



