QUADRATIC VECTORS. 407 



— Sy(t)'Y = — Sv(f)v = ~— S\pFp, 



dhy 



the second derivative of the ternary form SXpFp along the normal, 

 more conveniently written S^yV-S\pFp. By the commutative 

 property of S^yV we thus have 



Sa<Pa = a'^SXS'-yV-pFp, (111) 



These results substituted in (104) give, as the greatest curvature of a 

 normal section of the cone at a point where the normal is in the 

 direction y, 



<y2v2 — S'^yV 

 3=^^ Ta '^' (112) 



where the factor 7^ is introduced for homogeneity, in order that 7 

 need not be a unit vector. This new numerator thus defines a differ- 

 ential operation upon VpFp. 



We may now introduce the conditions for a triple axis. First, a is, 

 in direction, independent of X, hence is of the form ySr]\, when P, 

 the axis, is written for p. Therefore 



Ta = TySrjX, (114) 



and, in order that X may cancel from the expression for g, it is necessary 

 and sufficient that 



{y'V' - S'yV)VpFp 



shall be parallel to t] when /3 is put for p after the differentiation. 



But this condition may be still further simplified. Let a be the 

 direction which Fp7 takes when (3 is written for p, so that a, 13, and 

 7 form a rectangular system. Therefore, if they are taken of unit 

 length, 



^ V2 = -S^aV - S^iSV - SVV 



while S^/3V vanishes if (8 be put for p after the differentiation, because 

 /3 is an axis. This gives 



^2y2 _ 52^y ^ +S2aV, 



