QUADRATIC VECTORS. 



381 



Equations (15) therefore determine the three quantities (45ai), 

 (45a2), and (45a3), in terms of a constant of proportionality, which I 

 shall denote by ks, as follows, 



/4.„W7. |(124)(234), (234) (314) 

 (,^oai; /.6 I ^^25) (235), (235) (315) 



f45a,)-ii:J(2^^^(^l^)' (314) (124) 

 ^ '^ ' I (235) (315), (315) (125) 



(4503) = ^6 



(314) (124), (124) (234) 



(315) (125), (125) (235) 



= /.•6(123) (234) (235) (415), (18) 

 = Z;6(123) (314) (315) (425), (19) 

 = ^,-6(123) (124) (125) (435), (20) 



These equations are changed into one another by cyclic advancement 

 of the numbers 1, 2, and 3. By advancing cyclically the numbers 

 4, 5, 6 we may obtain two other sets, of three equations each, sufficient 

 to determine the quantities 



(56a 1), (5602), (5603); and (64ai), ^64a2), ((Has); 



respectively in terms of two other constants of proportionality ku 

 and ^5. 



These nine relations enable us to write {Fo{p) in terms of jSi, ^o. . ./Se 

 and the constants ^"4, /*5, A'e, by means of a vector equation. Con- 

 sider the determinant of the coefficients of the three vectors ^i, 85, 

 and Fo{p), which we may abbreviate (45Fop)- By (14) we have 



(123)2 (45Fop) = (45ai) (31 p) (12p) + (45a2) (12p) (23p) 



+ (45a2) (23p), (31p), (21) 



-whence, comparing with (15), we have 



■(123)2 (45/rop) = k. 



If we agree to write 



(31p) (12p), (12p) (23p), (23p) (31p) 



(314) (124), (124) (234), (234) (314) 



(315) (125), (125) (235), (235) (315) 



(22) 



P(p) = i (31p) (12p) + i (12p) (23p) + /.• (23p) (31p), (23) 



"We may conveniently denote the determinant on the right of (22) as 

 (PiP^Pp), since it is the determinant of the coefficients of the three 

 vectors Pi0i), P(fib) and P(p). By advancing the numbers 4, 5, and 

 '6 we obtain two similar relations and have 



