288 



WILSON AND MOORE. 



As the X's thus form a basis for the expression of systems in general 

 \we may set up readily the progressive product of Grassmann; for 



Y Y 



— ^ijCiC-j 



iK s yX s 



, etc. 



In the system of Grassmann the progressive product represents the 

 space determined by the elements (a parallelogram in the case of two 

 vectors); but the interpretation here is not so direct because the 

 systems Xr of the first order are not components of a vector, — they 

 have to be multiplied by certain factors to obtain components of a 

 vector. ^^ In like manner the terms XrYg — XaYr are not components 

 of a plane but may be converted into such by proper factors. 



13. Orthogonal unit n- tuples. We may define orthogonality 

 relative to a gi\'en quadratic form as in non-euclidean geometry. We 

 shall now however take the form as differential, namely, as 



Since the elements f/.r, form a contravariant system (§4) a direction 

 in space may be defined by any contravariant system X'""^ if we set 

 up the simultaneous differential equations 







dxz 



dXn _ 



X(n) ' 



(24) 



and it is in this way that the contravariant systems used above, and 

 pre\'iously defined as contravariant systems, are associated with 

 special directions. 



If we have two systems jX''^' , yX^'^ we define as is customary in differ- 



17 It is shown by Ricci and Levi-Civita {Math. Ann., 60) that if two dual 

 systems of the first order Xr, X''") are divided by y/ Orr and y] a("\ the resulting 

 expressions X,/ yj Orr, X(''V Va('''') may be regarded respectively as the orthog- 

 onal projections of one and the same vector upon the tangents to the coordi- 

 nate lines Xr and upon the normals to the coordinate surfaces; whereas the 

 expressions X(''Warr and Xr^a^") represent respectively the components of 

 the same vector along the same lines and the same normals. This process 

 of rendering a system vectorial might be called vectorization and could be 

 extended to vectors of higher order (Stufe). 



