QUADRATIC VECTORS. 443 



A simple solution is found by multiplying (216) by 2, and writing 

 b = A 13, and c = .4 12. The resulting quadratic vector is 



2(^13/32X2- + .412183X3-) - p(^13X2 + ^12X3) 



which by taking p = Xi/Si + X2i82 + X3/33 becomes 



i3i(— ^13X1X2— ^12X1X3) + 182(^13X2^—^12X2X3) + 183(^12X3^—^13X2X3), 



(217) 



which, by Art. 4, has the same axes as (216). If we take 



4>p = X2F/3l^2 + X3TW1, ) (^.^s 



dp= (.4,2X3-^l3X2)Fi82/33+Xi(.li2r/3i/32-.4i3F/33l8i), i ^'^ ""^ 



we shall have (217) identically equal to Vcppdp, aside from the scalar 

 factor S|8i|32i33. 



This completes the factorization of (165) for all possible cases. 



It was shown in Art. (27) that when the quadratic vector has a 

 triple axis, (besides the quadruple axis with vanishing polar vector), 

 it can be thrown into the form (170), evidently of constant plane. If, 

 however, we subtract the term 



p(^12 — A2l)Xi + P/I13X2 



the resulting quadratic vector may be written 



— X22^i3i32— X2X3(yli2i32— yl2ii32+^i3i83) + X32(au8i+a2|82 — ^12183 



+^21^3) (219) 



which is the same form as (214) and can be factored as in (215) pro- 

 vided the vector coefficients are diplanar. We cannot have .4x3 = 0. 

 The condition of coplanarity for the three coefficients is therefore 

 ai = 0; that is, the element (82 is an inflectional element of the cones 

 (3). If such is the case, we may, instead, subtract from (170) the 

 term p(yli2 — ^2i)x3. The resulting quadratic vector may be written 



XlX2^13/3l4-X32(ai^l+a2|32-^12]83+^2l/33)-X2X3(^12-^2l)/32 (220) 



which is in the form (203) and can be factored into Vcppdp if the vector 

 coefficients are diplanar; that is, unless 



^12 — ^21 = 0. 



