QUADRATIC VECTORS. 443 



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

 b = Ai3, and c = An. The resulting quadratic vector is 



2(^i3|S2a-2^ + A 12^ 3X3") — pi A 13X2 + A 12X3) 



which by taking p = .Tij3i + •J'2182 + 2^3183 becomes 



/3l(—^l3.^' 1-1^2— ^li2.ria;3) + l^oiA 13X2^— A 12X2X3) + /33(^i2a-3^— ^i3a^2.T3), 



(217) 



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



(/,p = .T2F/3i^2 + X3Vl33^u I ,21 Q^ 



ep= iAnX3-Ai3X-2}l%^3-{-xMy2V^,^2-AnV^3^0.S ^ ^ 



we shall have (217) identically equal to V(^pdp, aside from the scalar 

 factor Si3i/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 — A2^X3 + p^i3a;2 

 the resulting quadratic vector may be written 



—.r2^^i3i32—.T2a;3(Ji2/32— ^21182+-'! 13/33) + xi{ai^i-\-a2^2 — Ai2^3 



+^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 -4 13 = 0. 

 The condition of coplanarity for the three coefficients is therefore 

 a\ = 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(^i2 — A2\)x3. The resulting quadratic vector may be written 



a:i.r2^i3^i+a:32(ai^i+a2/32-.4i2^3+^2i|83)-a:2.r3(^i2-^2i)/32 (220) 



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

 coefficients are diplanar; that is, unless 



A12 — A21 = 0. 



