SIMILAR TO THOSE OF HYDRODYNAMICS. ' 381 



jm*- p—2n -.-.jr k—2n , ,,, p=2rc ,,,_ . 



rflf„ . V. _ dM p , ^ ( dM n , ^ T dJ\L ) , 



u=^+ 2 W + i2 2?+2^s£ +o (80) 



p=l *=0 ( * p=l * ) 



Equations (59), which correspond to the case of steady motion, give us, on multiplica- 

 tion by j4 i _ 11 , -4i_i, 2 > • • • • Ai_ Xn and addition of the results, 



1 / , dW , dW , dW\ 



*• =^(^-n ^ + ^-i.i *?+■••■ + 4_i.. ^r) ( 81 ) 



expressions similar to (56) for the determination of the functions <f>. 

 We have seen that if we have 



W = ? (82) 



then k is an integral of equations (73) or of 



dxx dx a dx n 



W = u i> W — "*""!* = u « (83) 



In order now to obtain an idea of the nature of the functions 6,6 1 . . . . n _i, 

 consider a function of the n + 1 variables x x , x 2 . . . . x n<t , which denote by f, i. e. 



f=f( Xl ,x % .... x n>t ). 



This function/ which satisfies say 



/ = const. 



sustains the same relation to the space of n dimensions as a function of the three 

 variables x, y, z, and also of t sustains to ordinary space, so for want of a better term 

 I shall speak of it as a hypersurface. Now suppose the hypersurface/to be composed 

 always of the same particles, it must obviously satisfy the differential equation 



It + "i d7, + "2 ax 2 + • ■ • • + «■ dj n = ° (84) 



(84) has as we know n independent integrals which we may denote by 



6, 0i, 2 ■ ■ ■ . n -i 

 The equations 



= const., #i = const., etc. 



are, therefore, the equations of moving hypersurfaces which contain at all times the 

 same fluid particles. The particles which are at any time on the intersection of any 

 n — 1 of these hypersurfaces will remain on that intersection. As before we denote 

 the Jacobian of the functions 0, $ 1 , 2 , . . . . #„_-, by J. So far as it has been con- 

 vol. x. 49 



