444 HITCHCOCK. 



If such is the case, and if, also, ai = 0, the vector (170) takes the 

 simple form 



^iAizXiX2 + jSsaaors^ (221) 



If we subtract the term pxz "^02^13, the resulting quadratic vector may 

 be written 



^i(AuXiX2 — hxiXs) + ^i{a2X^ — 6x2X3) — ^sbxs^ (222) 



where b = "^02^13. If we take 



4>P= ^a2X,V0203-\-_^AnXiVl338u 



dp = ( Vao .T3 - ^^ 13 ^2) ¥^1^2 + ^A 13 X3F^3i8i, (223) 



we shall have (222) identically equal to V(})pdp aside from the scalar 

 factor S^i^ilSs. 



We have thus considered all possibilities for an axis /3i when the 

 cones (3) all have a double element at j8i, and /3i is not of higher than 

 the fovu-th order. If /3i is of fifth or higher order we may assume 

 ^2 and 183 any two vectors not coplanar with jSi. By virtue of the 

 vanishing polar vector we may then throw the quadratic vector into 

 the form 



a 1X2^ + 02X2X3 + 03X3^, (224) 



and if the vectors a are diplanar we may factor as in (214) and (215). 

 If the as are coplanar, jSi must lie in their plane, for if not we shall 

 have three other axes in that plane, (not necessarily distinct), and /?i 

 will be of the fourth order only. We may suppose ^2, as yet arbitrary, 

 to be some other vector in this plane. (224) may then be written 



^l{BnX2^+Bi2X2X3+BizXi')+^2{B2lX2''-\-B22X2X3-\-B2zXz^), (225) 



where the B's are constant scalars. The plane of /3i and ^2 is the 

 same as the plane X3 = 0. It is e\adent, therefore, that the only 

 axis, if any, which (225) possesses distinct from |3i is 



^1^11 + M21. (226) 



If ^21 is not zero we may suppose ^2, as yet arbitrary in the plane 

 Xa = 0, to be an axis; when ^u will disappear. By subtracting the 

 term PB21X2, the resulting quadratic vector may be written 



