JoLY — Quaternion Invariants of Linear Vector Functions, 8jC. 15 



But the equation of this cone may he written in either of the forms 

 Scf}f^aVp<J3i-^(j)2p = 0, or Scl>n-^aVpcfif'^(fi^p = ; 



and therefore the axes of <^r^<^2 (or of its reciprocal (j>2~^cjii) also lie 

 on it, and are the three residual lines in "which it cuts the cuhic. 

 More generally, the equations of the quadric cones comhine into 



Sa Vp (fir^i + h(J3o + <?) p = 0, and Sa Vp {a'cf^i + b'<^2 + c') p = ; 



and, as before, their three lines of common intersection with the cubic 

 lie on the cone 



^a (ff^i + 5^2 + f)p («'0i + ^'4^2 + c')p - 0. 



This equation, being thrown into the form 



S {acpi + 14>2 + c)"^a Vp (ff<^i + 502 + cy- {a'4>i + J'^o + c')p = 0, 



shows that the axes of all the functions of the type 



(«<^i + hcf)2 + c)-\a'<^i + b'4>2 + c') 



are co-residual triads on the cubic cone. 



