428 HITCHCOCK. 



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

 Multiplying the first by /Ss and the second by (84 and taking scalars 

 we obtain two equations, 



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

 (457r)C5 + (451) (315) (125)w + (452) (315)2a = 0, ^^^^^ 



homogeneous in the three unknoAvns (457r), u, 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 jSi, 184, and 

 /Ss are coplanar, and we may put 



185 = m^i 4- 71/35. 



The matrix of the coefficients becomes 



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



mn{231) (314) + n^C,, 0, mn^ (412) (314)^, 



The axes 184 and 185 being assumed distinct, neither m nor n is zero. 

 We cannot have (412) = for the four axes |8i, 182, I3i, jSs, 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 



(3, = ?»|82 + n^i. 



The matrix of the coefficients becomes 



C4, m(421)(314) (124), ■ • 0, 



W7i(312) {(234) + f/i (124)} + n'C\, in{42l) {m(312) + 7i(314)}n(214),0, 

 and the only non-vanishing determinant is seen to equal 



- ^m2n(124)4(123) 

 no factor of which can vanish under the hypotheses. 



