QUADRATIC VECTORS. 433 



(^3 - sro)r/32/3_3 = ai, (gi - g3)Vl3s^i = a., {go - gi)V^i^, = a,, (200) 

 we find identically 



Vcfypdp = a1.T2.r3 + aoX^Xi + 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 = hiXiVaoas + hoXiVasai + h^x^Vaiai, 

 dp = CiXiFa2a3 + C2.T2l^ci3ai + CzX3Vaia2, 



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' Faia2Fa3ai = aiSaia2a3 



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

 three relations to determine the six constants, 



A3C2 — ^2C3 = hiCs — hsCi = hiCi —hiC2 = (202) 



oaia.2Cis 



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

 down <j)p and 6p. For example, a simple, although unsymmetrical, 

 solution, is 



^1 =1, A2 = — 1, ^3 = 0, CiS(aia2a3) = — 1, C2 = 0, 



C3S(aia2a3) = +1. 



If we let the resulting values of cf) and 6, or any two we may construct 

 satisfying the conditions, be called </)o and do, then the new pair, 



<t) = W0O + v6o, 6 = ui<{)o + tJi^o, 



will also satisfy them, provided vvi — uiv = 1. It thus appears 

 that we have V({>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 vi, with the restriction as given. 



