392 HITCHCOCK. 



PART III. VECTORS WITH MULTIPLE AXES. 



11. I shall now suppose that a quadratic vector is given possessing 

 three known axes, distinct and diplanar. With the notation and 

 ideas of the first part of this paper, we may suppose the vector to be 

 thrown into the form (12), by the addition of a term pS8p, that is tp. 



Let the three vectors ai, ao, as be expressed in terms of the three 

 axes /3i, 02, jSa by identities like (25), e. g. 



a]S/3u32i33 = /3iS|82^3ai + /Ja-S/Js/^iai + jSsS/SiiSsai, (59) 



the scalar of the product of three vectors being, sign excepted, the 

 determinant of their components. If we adopt the notation 



^11 = ;; . A21 = 7; , A12 = , etc., (60) 



S/3i/3o/33 S0i02l3s S/3uS2/33 



we shall have Fo{p) in the form 



Fo{p) = 0i{AnX2Xz + A12X3X1 + ^13:^1X2) 



+ 02{A2lX2X3 + A22X3X1 + ^23^:1X2) 



+ 183(^31^:20:3 + A32X3X1 + ^33X1X2), (61) 



where the nine A's are constants to be determined. 



If jSi is a double axis, the cubic cones (3) have the same tangent 

 plane at the element /3i, or else have a double line at /3i. By taking 

 polars,^^ the vector VpF{p) gives 



VpFoi^i) + V0MA22XZ + A2ZX2) + r/3i/33(^32.i-3 + .433x2), (62) 



because X2 and X3 vanish when j3i is put for p. But Fo(|Si) vanishes. 

 Hence, that we should have, at most, one polar plane for the three 

 cubic cones (3), it is necessary and sufficient that the determinant 



-422, A23 

 A 32, A 33 



(63) 



11 That is, diflferentiating VpFp and putting 0i for p after the differentiation; 

 we have, as the polar vector, Vp^Fp + Fp ||8i(AiiX2'.r3 + -4 11X22-30 + • • ■ etc. | , 

 which, writing 0\ for p, and dropping accents, gives (62). 



