402 HITCHCOCK. 



(36p) = 0, the tangent planes to the cones (3) at the double axes. 

 Any other vector having the same axes and the same tangent planes 

 can differ from the above at most by a multiplicative constant and an 

 additive term pS8p. 



With regard to the case of three double axes (91) gives the most 

 general form of such a quadratic vector. For, by its method of deri- 

 vation, it is always possible whenever the three double axes are not 

 coplanar. But no irreducible vector can have three distinct multiple 

 axes in the same plane : a fact we would perhaps guess from the 

 stand-point of Art. 3, if the same ideas apply to vectors with multiple 

 axes; for the plane of the three double axes, taken twice, would con- 

 stitute a quadric cone. It is more conclusive to prove directl}''. The 

 following method of attack is, moreover, applicable to a variety of 

 cases. 



Let two axes of a quadratic vector be /3i and ^2- Let 183 be some 

 third vector, not necessarily an axis, but such that (123) does not 

 vanish. Let a suitable term pSdp be added to the vector so as to make 

 zeros of /Si and (32, (by two equations like (8)). With the notation (9), 

 the resulting vector can have no terms in Xi^ or in a:2^ but may have 

 terms in Xs^. If we now expand as in (61), writing Bi, B2, and ^3, for 

 the coefficients of Xz^, we shall have 



/3i(y4ii.T2.T3 + ^120^3.^1 + AnXlXo + BiX^^) 

 + i32(-421.X-2a;3 + A22XzXi + A2ZX1X2 + B2Xz^) 



+ ^z{AzyX2Xz + ^32X30:1 + AzzXxX2 + Bzxf). (92) 



This is evidently a form to which any quadratic vector with two known 

 axes can be reduced with ease. 



The necessary and sufficient conditions that this vector possess a 

 third axis coplanar with ^1 and ^2, but distinct, are 



-^33 = 0, ^13 not zero, ^23 not zero. (93) 



For directions in the plane of /3i and ^2, but not along /3i nor ^2, are 

 given by 



0:3 = 0, xi not zero, X2 not zero • (94) 



If Xz = 0, and (92) lies in the plane of /3i and /So, we must thus have 

 necessarily ^33 = 0. The direction of (92) is then 



181^13 + /32^23. (95) 



