QUADRATIC VECTORS. 417 



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

 ated 



(527) E'p 



^4 



(452) £^ 



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

 through /3i and 185, having its curvature at 182 determined by the 

 constant a. We now have for the Hmiting vahie of (151), 



3 ^'^^Ep,b -(527) £^5£^p+ (247)£^£V (^57)(Q,,QM 



^' (452) £^ ^ '^' (452) (ES)^ "^ "' (452) {Q,,QM ' 



(155) 



where, in the second term, E/p contains 187 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 

 (185 + t^4 -\- fh^i) instead of /Si and let t approach zero. W^hen t 

 approaches zero we have 



j^-^^^ Ep ^ (235)^(23p) (45p) + a(236) (425) (52p)=^ 

 E-j (235)2(237) (457) + a(236) (425) (527)^' 



the numerator being a quadric which vanishes on a cone through 

 jSs and /So 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 writing for /3i its new value we have terms containing the square 

 and higher powers of t. We thus find 



Urn -^ = **^^^--^ (235)2(154)J-_a(236) (425)^^(524) 

 '^ (£v') [(23'5)2(237) (457) + a(236) (425) (527)2p 



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

 plify (QiiQ^Qp) we have 



