QUADRATIC VECTORS. 379 



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

 Fo, and Zo, that is 



F^{p) = zA^o + jYo + JcZo, 



the change of coordinates may be analytically represented by 



Xo = X +.t(poX + qoy + ^0^) = au.T2.T3 + anXsXi + anXiX2 



Yo = Y + y(:poX + qoy + ros)= a21.T2.T3 + a22a;3a:i + a23.T1.T2 (10) 



Zo = Z -\- z(j)ox + qoy + roz) = asiX2X3 + a32.'r3.i^i + a33a:i.T2, 



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

 a\, a2, and a^ be defined by 



ai=ian-\-jan-\-ka3i, a2 = iaiz-\-jao2-{-ka32, 03 = fa 13+ 7*023+ ^'a33, (11) 



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



Foifi) = 010:23:3 + a2T3a:i + 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 jSi, ^2, and 183 has been made. In other words, any two 

 vectors, alike in having /3i, ^2, ^z 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 ^i, ^5, and 

 iSe may be axes of Fo{p) and therefore of F(p). If Foifii) is a scalar 

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

 04, /Ss and FoiPt) vanishes. Abbreviate this determinant by (45i^o4)- 

 Similarly (45i^o5) vanishes if Foifib) is parallel to iSs- Advancing 

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

 conditions 



(45F04) = 0, (56F05) = 0, (64Fo6) = 0, (.0-) 



i^5Fo,) = 0, (56Fo6) = 0, (64Fo4) = 0, ^ ^ 



They are also sufficient; 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 ^0(184) 

 shall, at the same time, lie in the plane of ^i, ^5, and the plane of jSe, 



