342 HITCHCOCK. 



We now have three coplanar sets given by (234) = (315) = (126) = 0. 

 We may not include /St in a fourth set if the quadratic vector is to be 

 irreducible and have no central axis. We must therefore take (456) 

 = which evidently implies the vanishing of the determinant 



0, A21, A31 

 An, 0, A32 



^13, ^423, 



(36) 



Six axes of the quadratic vector will now be along the lines of inter- 

 section of four planes. This fact suggests an expression for the vector 

 Fp which shall be symmetrical in these six axes: for consider the 

 vector T'7i72S73piS74P, where the 7's are taken at right angles to the 

 four planes respectively; all six lines of intersection of the four planes 

 with each other are axes of this vector term, and the same will be true 

 no matter in what order we write the four subscripts. If, then, 

 «i2, Ois, 023, Oi4, 024, and 034 be any six scalars, the vector sum 



Fp = ai2F7i72S73P<S74P + «i3r7iT3'S72P»S74P -{ 1- a34l-''7374<S7ipS72P 



(37) 



will be a quadratic vector of the type under consideration, namely 

 it will have six axes in the directions T'7172, T"7i73, • • • F7374. This 

 form of Fp has a number of interesting properties easily proved by 

 methods already exemplified. As instances, — 



1. If pS8p be added, there are four values of 5 which render Fp 

 a binomial. One value is 



61 = - [023(123)74 + 024(124)73 + 034(134)72] (38) 



and the others are of similar form. 



2. If 5], 62, 63, 64 be the values of 5 just mentioned, the differences 

 5i — 82, 61 — 83, 82 — 83 etc. are all perpendicular to the seventh axis. 

 Hence the direction of this axis is easily calculated. 



3. The quadratic vector (37) possess some properties closely 

 analogous to those of the general linear vector function; for the 

 latter may be wTitten 



g23V}^2>^3S\ip + 5'3iT X3X1SX2P + gnVXiX^SXsP (39) 



and has for axes FX2X3, etc. 



4. If Fp has the form (37) the cubic vector T'pFp takes the form 

 26i7iS72pS73P)S74p where 61 = — 012 — 013 — an and bo, 63, &4 are 

 easily found. The sum of the 6's is zero. 



