428 



HITCHCOCK. 



two vector equations equivalent, in general, to four scalar equations. 

 Multiplying the first by 185 and the second by ^i and taking scalars 

 we obtain two equations, 



(457r)C4 + (451) (314) (124)w + (452) (314)2a = 

 (457r)C5 + (451) (315) (125)u + (452) (315)% = 0, 



(188) 



homogeneous in the three unknowns (457r), 71, and a. The two-row 

 determinants from the coefficients cannot all vanish if Fp is not re- 

 ducible. For the determinant of the second and third columns is 



(451) (452) (314) (315) { (124) 315) - (125) (314) } 



which by a transformation already used becomes the product of de- 

 terminants 



- (451)2(452) (314) (315) (123). (189) 



Considering these factors in order, if (451) = 0, the axes /3i, ^4, and 

 /Ss are coplanar, and we may put 



185 = -"'jSi + "185. 

 The matrix of the coefficients becomes 



Ci, 0, m(412) (314)2, 



mw(231) (314) + n^C, 0, vm'^ (412) (314)^, 



The axes ^4 and /Ss being assumed distinct, neither m nor n is zero. 

 We cannot have (412) = for the four axes jSi, ^2, ^i, ^5, would be 

 coplanar and Fp would be reducible. We cannot have (314) = 

 for we cannot have two distinct axes in the tangent plane to (3), viz. 

 (31p) = 0. And we have (123) different from zero by hypothesis. 

 Hence this matrix cannot have its rank reduced to one. 



Taking the second factor of (189), if (452) = 0, the three single 

 axes are coplanar. We may put 



1S.5 = m^2 + n^i. 



The matrix of the coefficients becomes 



C4, m(421)(314) (124), 0, 



7;m(312) {(234) + g^ (124)} + Ji'd, m(421) {m(312) + 7i(314)}n(214),0, 

 and the only non-vanishing determinant is seen to equal 



- gvMl2Ay{l23) 

 no factor of which can vanish under the hypotheses. 



