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 t^'pe, we may write, 

 as in (89), 



Q{p) = i{12py + j{12p) (23p) + /.(23p) (31p), (150) 



giving ^2 a double axis, the tangent to (3) at /So being (23p) = 0. Put 

 Q for P in (88), with a = jS4. This gives, dropping the constant kr, 



^' (456) {Q,Q,Q7) (456) {QoQ.QiY 



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 do is (45p) = 0. The single axes are jSi, jSe and jS?. Let 

 /Se be replaced by ^2 + t^s + at^^e, and let t approach zero. Qe 

 takes the form Q2 + /(?23 + ^"(^3 + ciQ^e) plus terms containing higher 

 powers of /. But Q2 and Q23 vanish identically. xVlso 



Q^ = i{V23Y, Q2, = A-(123) (236), (152) 



so that the determinants, (or scalar products), (QbQsQp) and {QbQ^eQp') 

 give on expanding 



(Q5Q3QP) = (123)2(235). (23p) (15p), (153) 



(Q.QnQp) = (123)2(236) (125) (12p) (52p) (154) 



We have now merely to write Q3 + 0(^26 instead of Qa in (151). The 

 iirst term of (151) becomes 



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

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



