QUADRATIC VECTORS. 



417 



which by a notation in keeping with that already used may be abbrevi- 

 ated 



(527) E'p 



'^^ (452) E^^ 



where E^p denotes a quadratic form whose vanishing defines the cone 

 through /3i and 185, having its curvature at ^2 determined by the 

 constant a. We now have for the limiting value of (151), 



(527)£^ -(527)£^45£ V+ (247)£^£V {^^){Q,,qM 



^' (452) ES^^' (452) {ESY (452) {q,^M ' 



(155) 



where, in the second term, E'^p contains /St instead of /Ss. This vec- 

 tor is also of a type previously examined, viz. it has one triple and 

 one double axis. If the determinant (527) = it becomes a bi- 

 nomial, in agreement with the vector (147). We have now to write 

 ((85 + 04 + i^b^i) instead of /3i and let t approach zero. When t 

 approaches zero we have 



Ep ^ (235)n23p) (45p) + a(236) (425) (52p)^ .^^g. 

 '"' E] (235)2(237) (457) + a(236) (425) (527)^' 



the numerator being a quadric which vanishes on a cone through 

 05 and 182 with tangent planes at those elements respectively (45p) = 

 and (23p) = 0, and having the constant a arbitrary. 

 Again, E''p does not vanish in the limit, but becomes 



(523) (237) (23p) (57p) + a(236) (527) (52p) (72p), 



obtained by writing 7 for 5 and 5 for 1 in E^p. 

 Now £^45 may be expanded as 



E45 = (123) (235)2(154) + a(236) (125)^(524), 



and on wi'iting for |8i its new value we have terms containing the square 

 and higher powers of t. We thus find 



J. . Eh^ ^ 6(523 ) (235)2(154) + a{23&) (425)' (524) 

 ^'"'^ (Ev^) ~ [(235)2(237) (457) + a(236) (425) (527)2]2 



Considering finally the third term of (155), if we expand and sim- 

 plify {QibQhQp) we have 



