JoLY — Quaternion Invariants of Linear Vector Functions, 8fc. 11 



15. Locus of axes of (pi + tcjiz- — ^ few properties of the system of 

 linear vector ftmctions <^i + ^^3 may here be noticed. If g^, g-i, and g-^ 

 are the roots, and pi, p2) ^'^^ Rs the axes corresponding to a given value 

 of t, the vector equation 



4>iP + i4'2P = 9P 



is satisfied for p = pi, and g = gi,'&c. Eemembering that <7i is a root 

 of the cubic 



g^ - /»h (<^i + tcjio) + gnh (^1 + tcfiz) - Ms (<^i + t(f)2) = 0, 



the vector equation denotes a cubic cone, the locus of axes of <^i + tcfit ; 

 the scalar equation of this cone is 



Sp<liip(f)2P = =/. 



The locus of axes of the conjugate system ^i' + t<p2 is given by 

 <kip + t^2p=gp, or by Spcji^'pcfi^'p = =/'. 



If pi is the edge of the cone / ', determined by t and gi, it is at right 

 angles to two edges p^ and p^ of the cone / which correspond to the 

 same value of t ; the vector Fp/^/p/ or Vpicfi^'pi is the third edge of/ 

 at right angles to p/.^' 



Thus the reciprocal of the cone /' is the envelope of the principal 

 planes of the system of functions (jti + tcji^. This envelope being of 

 the third class, through any line through the origin it is possible to 

 draw three principal planes of the system of functions ^1 + t<f)2. 



Equating to zero the discriminant of the cubic in g, it appears 

 that six functions of the system have double roots and coincident 

 axes ; for this discriminant is a sextic in t. The cone / being of the 

 sixth class, and the reciprocal of /' of the third, eighteen principal 

 planes of the system <^i + ^^2 are tangent to /, as well as to the reci- 

 procal of /'. Six of these planes evidently touch / along the six 

 coincident axes, and the other twelve planes probably correspond to 

 coincidence of p2 (or of ps) with Fp/<^ip/. 



The planes joining corresponding edges of the two cones (pi, pi ; 

 P2, P2'; and p3. Pa'), which answer to the same value of t, intersect on 

 the orthocentric line 



F(ei + ^e,)(<^i + ^«^2)(ei + te^). 

 As t varies, this line describes a cubic cone. 



* A particular case of this cone was considered in Art. 15 of my Paper on 

 "The Scalar Invariants of Two Linear Vector Functions" {loc. eit., p. 721). 



