QUADRATIC VECTORS. 413 



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

 fore not contain jSs or /Se so long as the quadratic vector Fp is not 

 reducible. 



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

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



VpB. (Me_^-^P') + VpB. mtJAPuPpp;) ^ y. miiPuPsPpp') 

 ^^' (P5P6P14) (156)Z)6 -(156)1)5 



+ Vp'Fp, 



where Fp has the form (136). Writing /3i for p, the first and fourth 

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



VPuPiP = t"(123)3 (41p), (138) 



whence, by (23), 



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



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



The vector x, that is the vector coefficient of /3i, is thus 



(614) (3 16 ) (126) _ (154) (315) (125) ,,,„ 



^' (T^D, "' (156)D, ' ^""^ 



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

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



(156)1)6 (156)D5 



which is VjSiT. 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 



Z>5 = (P5P14P4) + c(315) (125) (123)3 (417), 



De = (P6P14P4) + c(316) (126) (123)3 (417). (142) 



The determinant /SttjSit then vanishes identically, and the test is 

 completed. 



Any irreducible quadratic vector, having j3i a triple axis, and four 

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



