SIMILAR TO THOSE OF HYDRODYNAMICS. 369 



we may consider them combined in groups of three each ; i. e. consider the motion in 



n (n - 1) (n — 2) 

 6 



flat spaces of three dimensions instead of the motion in one space of n dimensions. 

 Equation (7) shows the relation that exists between the rotation components &. in 

 such a space ; of course this relation holds for all of the Euclidean spaces which can 

 be formed from the space of n dimensions, and therefore a similar relation exists for 

 the n-dimensional space formed by suming the expression on the left-hand side of 

 (7) for all values of I, m, n. Write for brevity 



n(n-l)(n- 2) _ 



6 " 1 - 



If the motion in any one of the I three dimensional spaces is rotational, it is of 

 course rotational for the space of n dimensions. 

 For the space containing x h x m , x n , we have 



(25) 



where y 2 is written for 



For ordinary space the quantities in parentheses vanish and these reduce to well- 

 known forms. Write them in the forms 



(26) 



Form now the quantities 



*'=£-£ ^ = etc - < 27 ) 



These operations cause the quantity R to disappear and leave us 



