438 HITCHCOCK. 



axes are in the plane of the vectors ai, a2, as. But /3i must He in that 

 plane by reason of its triple character. This is contrary to the hypo- 

 thesis that /3i and the two axes are diplanar. 



Case 2. The quadrics are not tangent at /3i. One of the two single 

 axes not selected to be zeros must be a zero, since the quadrics have 

 four intersections. But the plane of the vectors a contains three 

 axes, is therefore the tangent plane to (3) at jSi, and passes through the 

 non-zero axis, contrary to hypothesis. 



A similar proof can be obtained analytically by considering the con- 

 dition of Art. 19 that /3i shall be a triple axis. 



Under case 2 use was made of the fact that, if the plane of a quad- 

 ratic vector is constant, and if two of the three axes which it must 

 then possess in that plane are coincident, the plane is tangent to 

 the cones (3) at the double axis,— assuming this tangent plane to be 

 uniquely determined. An analytical proof of this is desirable. Sup- 

 pose first that the double axis is a zero, as in the case of the triple 

 axis just considered. Let coordinate axes be taken so that z is the 

 zero, j also in the constant plane, so that the vector becomes 



i(b.xy + cy^ + terms in 2) + j(,bixy + Ciy- + terms in 2). 



The equation to determine axes in the plane 2=0 then becomes 



x{bixy + ci?/-) = y{bxy + c/). 



That there may be only one axis other than y =Owe must have 61= 0. 

 But the polar vector of VpFp at i takes the form 



^{biy' + a term in 2'). 



Hence the condition that the tangent plane to (3) shall be the plane 

 2 = is also that we have 61 = 0. 



If the double axis is not a zero, we shall have the vector in the form 



i(ax^ + bxy + c?/^ + terms in 2) + jibixy + Ci?/^ + terms in 2), 



where i is taken as an axis. The equation to determine axes in the 

 plane 2=0 becomes ■ 



x{bixy + ciif) = yiax"^ + bxy + cy") 



giving 61 = a if there is to be but one axis other than i. But the 

 polar vector becomes at i, 



V{ix' + ji/ + kzjia + Vi{jbiy' + a term in z'). 



