416 HITCHCOCK. 



planar. The left members of (149) have no common factor. Thus 

 all the conditions are satisfied for the case in question. 



23, The case of a vector having two triple axes is of particular 

 interest as affording the first example of a quadratic vector which 

 cannot always be written in the form (61). For the triple axes and the 

 single axis may be coplanar. To build up this type, we may write, 

 as in (89), 



Q{p) = ^(12p)2 + i(12p) (23p) + ;c(23p) (31p), (150) 



giving /So a double axis, the tangent to (3) at 182 being (23p) = 0. Put 

 Q for P in (88), with a = ^i. This gives, dropping the constant k^, 



^' (456) (Q5Q6Q7) (456) {Q,Q,Q,y 



{4:5 7){Q,,Q ,Qp) . 



which is of a type previously considered, viz. it is an example of the 

 most general quadratic vector having two double axes. The tangent 

 to (3) at Bt is (45p) = 0. The single axes are |Si, jSe and ^7. Let 

 06 be replaced by 182 + t^z + af^^, and let t approach zero. Qe 

 takes the form Q2 + ^^23 + f^iQs + 0^26) plus terms containing higher 

 powers of t. But Qo and Q23 vanish identically. Also 



^3 = t(123)2, ^26 = A:(123) (236), (152) 



so that the determinants, (or scalar products), (QbQsQp) and (QiQiiQp) 

 give on expanding 



(Q.QzQp) = (123)2(235) (23p) (15p), (153) 



(Q5Q26QP) = (123)2(236) (125) (12p) (52p) (154) 



We have now merely to write Qz + ^§26 instead of Qe in (151). The 

 first term of (151) becomes 



(527)[(123) (235) (23p) (15p) + a(236) (125) (12p) (52p)] 

 ^' (452) [(123) (235) (237) (157) + a(236) (125) (127) (527)'] 



