QUADRATIC VECTORS. 433 



(^3 - f72)I W3 = a,, (g, - </3)r/33/3i = a.,, (g^ - gi)V%^. = a^, (200) 

 we find identically 



V<t)p6p = aiOToa^s + a2.V3-Vi + 03X1X2. (201) 



It has already been shown that any quadratic vector of the general 

 type, or any having three distinct diplanar axes, can be thrown into 

 the form of the right member, by means of the addition of a properly 

 chosen term pS8p. We have now to examine the converse of the 

 process by which (201) was just obtained, viz. to convert the right 

 member into the left, or to factor, vectorially, into the linear vectors 

 </)p and dp. In the most general form of quadratic vector this will be 

 possible. For we may write 



(f)p = hiXiVaias + hiXoVasai + h3X3Vaia2, 

 dp = CiXiVaoas + C2X2Va3ai + C3X3Faia2, 



where the six constants h and c are undetermined. If we take the 

 vector product of corresponding members, utilizing the identity, 

 proved in all works on vectorial algebra, 



V- Vaia2Va3ai = a 18010203 



with two others of like form, and compare with (201), we find these 

 three relations to determine the six constants, 



^3C2 — ^2^3 = hiC3 — hzCi = ^2Cl — ^lC2 = ^ (202) 



0010203 



whence, evidently, there are an infinite number of ways to write 

 down </)p and dp. For example, a simple, although unsymmetrical, 

 solution, is 



hi =1, ^2 = — 1, ^3 = 0, CiiS(oi0203) = — 1, C2 = 0, 



038(010203) = +1. 



If we let the resulting values of <l> and 6, or any two we may construct 

 satisfying the conditions, be called 4>o and da, then the new pair, 



will also satisfy them, provided uvi — UiV = \. It thus appears 

 that we have V4>pdp determinable by fifteen scalars ; each linear vector 

 in general involving nine scalars, but in the vector product we have 

 the four parameters u, v, ui v\, with the restriction as given. 



