444 HITCHCOCK. 



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

 simple form 



i8i^i3.TiX2 + ^3202X32 (221) 



If we subtract the term pxs "^02-413, the resulting quadratic vector may 

 be written 



/3i(^i3XiX2 — bx^xz) + fiiia^xi — hx^xz) — ^zhxz^, (222) 



where h = '^aoAiz. If we take 



</)p = ^a2^xzV^2^3 ±_^An xiV^zQu 



dp = ( Vflo xz - ^A 13 X2) F|8u82 + ^A 13 xzV(3z^u (223) 



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

 factor S^^2^3. 



We have thus considered all possibilities for an axis jSi when the 

 cones (3) all have a double element at |8i, and 13 1 is not of higher than 

 the fourth order. If /3i is of fifth or higher order we may assume 

 182 and /Ss 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^ + aiXiXz + a3.T3^, (224) 



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

 If the a's are coplanar, /3i must lie in their plane, for if not we shall 

 have three other axes in that plane, (not necessarily distinct), and j8i 

 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''-{-BnX2Xz+BlzXz^)+^2{B2lX2''-\-B22X2X3-{-B2zXz''), (225) 



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

 same as the plane Xi = 0. It is evident, therefore, that the only 

 axis, if any, which (225) possesses distinct from j3i is 



iSi^ii + ^2B2i. (226) 



If JB21 is not zero we may suppose 182, as yet arbitrary in the plane 

 Xz = 0, to be an axis; when J5u will disappear. By subtracting the 

 term pB 21X2, the resulting quadratic vector may be written 



