Jolt — Quaternion Invariants of Linear Vector Functions, 8fc. IS 



To prove these relations, observe that 



gzSp2^zp3 = Spi^ips = Sp3^ip2 = g^Spz^zpi ; 

 and therefore, as $1 and $3 are self-conjugate, 



Spz^sPz = Spz^spi = Spi^3p2 = 0, 



and 



Sp2^1p3 = Sp3^ipi = ^Pi$i/32 = 0, 



provided the roots ffi, ffn, and ^3 are unequal. 



It appears, therefore, that the axes of ^1 are a self -conjugate triad 

 of lines with respect to the two cones Sp^ip = and Sp^^p = 0. 



If, in addition, ^2 = ^3~^^2> the axes of ^2 would also be a self- 

 conjugate triad with respect to Sp^^p = ; but if two triads of lines 

 are self-conjugate to the same quadric cone, both triads must lie on a 

 quadric cone.^ The condition is, therefore, the axes of <^i and <^2 

 must lie on a quadric cone. 



Eeturning to the functions $1 and ^s, since 



Vpi^iPi = ffi Vpx^zpi = x^gy Vpi Vp2P3 1 1 (<^i - ffi) ei, 



the axes of <^i lie on the quadric cone /S Fp$ip Fei^iCi = 0. This cone 

 contains also the axes of <l>i, and, by a little manipulation, its equation 

 may be thrown into the form S Vp(j>ip Fki^jei = 0. In like manner, 

 the axes of $3 lie on the cone 



Srpcl,,pVe,^,e, = 0. 



18. Triads of lines in 2^ersjpective with their derived triads. — Gene- 

 rally the locus of lines p, any one of which is coplanar with its derived 

 line ^ip and a fixed vector a, is the quadi-ic cone Spc^ipa = 0. If this 

 cone contains a triad of mutually rectangular lines i, j\ and Tc, 



Sa%Vicj)-yi = 0, or Saey = 0. 



1 If the triangle, the coordinates of whose vertices are xi, yi, zi ; xz, yi, zz ; 

 and X3, ys, Z3, is self- conjugate with respect to 



ax- + by'^ + cz^ = 0; axzx^ + byzyz + cz^zz = 0, &c., 



= 0. 



and therefore 











1 



1 



1 





Xy 



yi 



\Z\ 





1 



1 



1 





X2 



2/2 



22 





1 



1 



1 





X3 



ys 



23 



