HI. QUADRATIC DIFFERENTIAL FORMS. GENERALIZED VECTORS. 565 



The geometric product of two displacements is equal to the sum of 

 products of the coordinates of either vector by the reduced components 

 of the other. Thus the geometric product is denned by means of the 

 quadratic differential form l) denning the space in question. 

 Solving the equations 3) for dq v dq%, we obtain 



6) 



from which we obtain 



7) ds 2 = E n dq^ -f 2 J2 12 dq dq 2 -f E 22 dq 2 2 . 



The expression 7) is called the reciprocal form to l). Corresponding to 

 it we obtain the form of the geometric product 



We may now define any vector belonging to the space considered, 

 as one whose components have the same properties as those possessed 

 by those of an infinitesimal displacement. Suppose that X, Y, Z are the 

 rectangular components of a vector E, it does not belong to the space l) 

 unless it is tangent to the surface in question. If so, we have a displace- 

 ment such that 



ds dx dy dz dq l dq^ s 



Then Q^ Q 2 are the coordinates of the vector in the system g A , <? 2 , and 

 the magnitude of the vector is given by the equations 



E 2 = X 2 + Y 2 + Z 2 = ~(dx 2 -f dy 2 + 



10) = ( Ed ^i 2 + 2Fd qi dq 2 -f 



where 



Ci 



% 



are its reduced components belonging to its coordinates Q^ Q%. The 

 geometric product of two vectors J2, It' is 



12) ii f + 8 a '-i'i+ 8 ' a - 



If now one of the vectors is finite, the other an infinitesimal displacement, 

 we have the geometric product 



