QUADRATIC VECTORS. 413 



an osculating cone to the cones (3) at the element /3i, and will there- 

 fore not contain 185 or ^^ so long as the quadratic vector Fp is not 

 reducible. 



As a verification, the test for a triple axis mav be applied to the vec- 

 tor (136) by the rule of Art. 18. The polar vector of VpFp is 



.. . (P,PePpp') .jr^ (614) jPePuPpp') , .. . (154) (PuP.Ppp') 



{P^PePii) (156)1)6 — (156)l)5 



+ Vp'Fp, 



where Fp has the form (136). Writing /Si for p, the first and fourth 

 terms vanish. To evaluate the other terms note that, by (86), 



VPuPiP = i(l23Y (41p), (138) 



Avhence, by (23), 



(PePuPip) = (316) (126) (123)3 (41p), 



(P14P5P1P) = - (315) (125) (123)3 (41p). (139) 



The vector x, that is the vector coefficient of jSi, is thus 



. (614) (3 16 ) (126) „ (154) (3 15) (125 ) ,,,_. 



^' (156)D, ^'-—(m)D. ' * ' 



common factors of the two terms being dropped. Writing a for p 

 and /3i for p' in the polar vector, the result is 



r,^. + V,,,, m<^^i^ + ,-,.,. mj^^, (14.) 



(156)L>6 (156)L'5 



which is F/3iT. The scalar product of (141) and (140) must vanish. 

 This verifies at once, by actual multiplication, the denominators 

 being transformed as in (138), so that 



D, = {P,P,,Pi) + c(315) (125) (123)3 (417), 



I>6 = (P^PuPi) + c(316) (126) (123)3 (417). (142) 



The determinant Stt^it then vanishes identically, and the test is 

 completed. 



Any irreducible quadratic vector, having /3i a triple axis, and four 

 -other distinct axes, may be thrown into the form (136). For the only 



