566 NOTES. 



Thus the reduced components ft, Q 2 are given by the definition which 

 we have adopted on page 116, equation 42). 



Having now illustrated the subject by a space of two dimensions, 

 we can easily extend our notions to space of any number of dimensions 

 m, defined by the form 



14) ds* 



For any one of the coordinate directions we have 



15) ds r * = Q rr dq r \ 



and for the geometric product of two displacements, 



16) dsds 1 cos (ds, ds') = ^ ^ s Q rs dq r dq s '. 



If one is in the direction ds r , all dq's being except dq r , 



17) dsds r cos (ds, ds r ) = ^. s Q rs dq r dq s , 



and dividing by ds r we obtain the orthogonal projection of ds on ds r 



18) d6 r = ds cos (ds, ds^ = ^_ g % 



VQrr d( lr 



from which we obtain by the definition of the reduced component 



da 



19) 



d r 



We have as before 



20) 



21) dsds' cos (ds, ds 1 ) 



and if the solution of 19) is 



22) 

 we have the reciprocal form 



23) ds* 



Again we may define a vector belonging to the hyperspace considered, 

 and now the rectangular components may be of any number, the limitation 

 of the vector to the space in question reducing the number of generalized 



