QUADRATIC VECTORS. 379 



in x^ nor in x^. If the original components of Fo(p) be called Xq, 

 Fo, and Zo, that is 



/"oCp) = iX^ + il'o + ^1^, 



the change of coordinates may be analytically represented by 



Zo = X +x(poa: + q^y + roz) = 0110:2X3 + 012X3X1 + 013X1X2 

 Y^= Y ■\- yijjQX + qoy + roz)= 021X2X3 + 022X3X1 + 023X1X0 (10) 

 Zo = Z + 2(j»oX + qoy + ros) = 031X2X3 + 032X3X1 + 033X1X2, 



where the nine o's are constants to be determined. If three vectors 

 a\, a2, and as be defined by 



ai=iOii+jo2i+^'03i, a2=iai2+ 7022+^*032, a3= {013+7023+^033, (11) 

 it is obvious that the transformation (10) is equivalent to 



Fo(p) = aiX2X3 + a2X3Xi + 03X1X2 (12) 



Taking this result as one step in the demonstration of theorem I, 

 we note that the form of the right member is determined when the 

 choice of axes /3i, ^2, and /33 has been made. In other words, any two 

 vectors, alike in having /3i, 182, fiz for axes can, by a proper choice of 

 _p, q, and r, be thrown into the form (12), and will then differ in the 

 vectors a but not otherwise. 



Consider next the disposition to be made of the a's that 184, 185, and 

 ^6 may be axes of Fo{p) and therefore of F{p). If Foifiij is a scalar 

 multiple of ^i, the determinant of the coefficients of the three vectors 

 Qi, /Sb and Fo(fii) vanishes. Abbreviate this determinant by (45i^o4)- 

 Similarly (45i^o5) vanishes if Foifi^) is parallel to /Ss. Advancing 

 cyclically the subscripts 4, 5, 6, we have in this manner six necessary 

 conditions 



(4.5F04) = 0, (56^5) = 0, ((MFoe) = 0, .,o) 



(45F05) = 0, (56Fo6) = 0, (64Fo4) = 0, ^ ^ 



They are also sufiicient; for, pairing the six relations in a different 



manner, 



(45F04) = 0^ (56F05) = 0, (64Fo6) = 0, 

 (64Fo4) = 0, (45F05) = 0, (56Fo6) = 0, 



we see that the two equations of the first column require that Fo{^i) 

 shall, at the same time, lie in the plane of 184, ^5, and the plane of /Se, 



