QUADRATIC VECTORS. 



381 



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

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

 shall denote by ke, as follows, 



(45ai) = Z;6 



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



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



= /C6(123) (234) (235) (415), (18) 



<«"=) = "' ' (235) IlU ill (IS) I = '"-'(l^S) (314) (315) (425), (19) 



(45a3) = ke 



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



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



= A;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 i), (56a2), (56as); and (64ai), i,64a2), (64a3); 



respectively in terms of two other constants of proportionality ki 

 and ki. 



These nine relations enable us to write (Fo{p) in terms of ^i, ^2- ■ /Se 

 and the constants ki, k-^, ka, by means of a vector equation. Con- 

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

 and Fo{p), which we may abbreviate (45/^op). By (14) we have 



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



+ (4.5a2) (23p), (31p), (21) 



"whence, comparing with (15), we have 



•(123)2 (45Fop) = 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) = { (31p) (12p) + j (12p) (23p) +■ k (23p) (31p), (23) 



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

 (P4P5PP), since it is the determinant of the coefficients of the three 

 vectors P(0i), P^Ss) and P(p). By advancing the numbers 4, 5, and 

 '6 we obtain two similar relations and have 



