III. QUADRATIC DIFFERENTIAL FORMS. GENERALIZED VECTORS. 563 



In this equation the afs are not all arbitrary, but as before & may be 

 determined as linear functions of the remainder, say a? 1 , . . x k in terms 

 of #_f.i, . . . flJ/n, which are arbitrary. But the multipliers A are as yet 

 arbitrary. Let us determine them so that they satisfy the equations 



A + *i #11 + *2 #21 ' ' + A* JBti = 0, 



v ^-2 ~l~ ^1 #12 ~ 



#22 * ' * -f ^/fc#&2 = 0, 



J-yfc -f ^i #1 A + Aj #2* ' ' * ~ 



which are just sufficient to determine them We thus have 



+ A t -Bl, yfc-fl + + 



5) 



+ (A m 4 A! 5 lm + - - + A 4 S km )x m = 0, 



in which the x's are all arbitrary, so that the m k coefficients must 

 vanish, giving the m If equations. 



= 



=0. 



Inserting in these the values of the I's already found, we have the 

 m Jc required relations between the A' 8 and J5's. Obviously the result 

 of the elimination may be expressed in the form obtained by writing 

 equal to zero each of the determinants of order fc + 1 obtained from the 

 array of A's and J3's in equations 4) and 5) by omitting m ft 1 

 rows, only m & of the determinants thus obtained being independent. 



NOTE III. 



QUADRATIC DIFFERENTIAL FORMS. GENERALIZED VECTORS. 



The method of transforming the equations of motion used in 37 

 and the application of hyperspace there occurring render a somewhat 

 more detailed treatment of the question desirable. In order to elucidate 

 matters, we will begin with the very simple case of a space which is 

 included in ordinary space, namely the space of two dimensions forming 

 the surface characterized by two coordinates ^n # as on P a e HO- 

 We have seen that this space is completely characterized by the expression 

 for the arc as the quadratic differential form 



A point lying on this surface may be displaced in any manner, in or 

 out of the surface. If it is displaced in the surface, its displacement is 

 a vector belonging to the two-dimensional space considered. We will 



36* 



