436 HITCHCOCK. 



Four of the seven axes are zeros, viz. the four intersections of 



2:2^3 + mziXo = 0. .TaXi + nxiXi = 0. 



There can be no other axis without the plane of ai and 02. For the 

 above pair of simultaneous equations cannot have five solutions since 

 the number is finite. No axis not given by a solution of these equa- 

 tions can be a zero. That is, the remaining axes are not zeros. Hence 

 they are in the plane of ai and a2. Since, by hypothesis for this case, 

 the three selected zeros are single axes, the fourth zero is single. Of 

 the remaining non-zero axes, one is therefore a double axis. In virtue 

 of the method by which the first three zeros were selected, the last 

 non-zero axis cannot lie in the tangent plane at the double axis, which, 

 therefore, cannot be the plane of ai and a^. 



Consider the normal form (88) ; which is precisely the present case, — 

 three single axes being zeros, the double axis being /Ss, and the tangent 

 plane at jSs passing through a, which by hypothesis, is distinct from /Ss. 

 If (88) lies in a constant plane, that plane is therefore the tangent 

 plane, contrary to the above result. Hence the vector cannot lie in a 

 constant plane; that is, ai, ao, as, cannot be coplanar. 



Case 2. If the three vectors selected do not remain single in the 

 limiting vector, let the double axis be |3i and the other be ^2 and let 

 the vector be thrown into the form (171), and ^2 be rendered a zero. 

 We then have 



Fp = 7rXi.T2 + TXi^ -\- ^X2Xz. (203) 



The vectors tt, t, and f will not be coplanar. 

 Proof. Suppose tt, t, f , to be coplanar. Put 



f = m-K + nr. 



The vector Fp takes the form 



irixiXi + mxiX^ + T{xi -f nX2Xz). 



By the same reasoning as in Case 1, the non-zero axes must lie in the 

 plane of tt and t. By virtue of the method of selecting the first three 

 zeros, if the non-zero axes are three single axes, they are not coplanar. 

 Hence they cannot be all single. One non-zero axis is therefore double. 

 By the same reasoning as in Case 1, the tangent plane at this latter 

 double axis cannot be that of tt, and t. 



Consider the model form obtained by wnriting Q for P in (88), 

 which is precisely the present case. As above, the vector, if in a 



