L 92 J 



VI, 



THE QUADRATIC VECTOE FUNCTION. 



By EEV. THOMAS ROCHE, M.A., B.Sc, B.Pn., B.C.L., 



St. Patrick's College, Mayuooth. 



Eeiid November 11. Published Decemuek 14, 1912. 



Introduction. 



A LINEAR vector function of a vector contains the independent variable to 

 the first degree. If we represent this variable by p, the function, being a 

 vector, may be written aSa'p + liS\5'p + ySy'p, a foi-m obtained by resohang 

 the vector along the three vectors a, /3, y. If a = a, /3 = jS', y = y, the 

 function is self-conjugate ; but it is not so in general. This is the most 

 general form of linear vector function of p. The properties of this function 

 have been examined very fully by Joly and Tait, following Hamilton. It can 

 always be inverted, that is, we can always find a solution of the equation 

 (T = 0p, where o- and form of are given, and p is to be determiaed. We 

 have written <^p for the general linear vector function. A very remarkable 

 property of this function is the following, proved in any treatise on Quater- 

 nions* : — Every linear vector satisfies a symbolic cubic equation of the form 



^ - m"(j)' + 7>i'(j) - m - 0. 



The meaning of this is that the operator on the left annihilates every 

 vector, (f)^ and <jt^ are used in their ordinary sense. The coefficients in this 

 cubic are called the invariants of the function. 



Thus the general linear vector function is easily investigated. It is not 

 so with the general quadratic vector function. This function contains the 

 independent variable to the second degree. It is a vector, and if we resolve 

 it along the three known vectors a,, a,, a,, the function takes the form 

 xai + ya, + zas, X, y, z being scalar quadratic functions of p, and therefore 

 may be expressed as Sp^^p, Spfip, Sprpp,, where <pi, <j),, <p3 are linear vector 



*!'. Joly, Manual of Quaternions, pp. 93, foil. 



