QUADIL^TIC VECTORS. 407 



(P 



— Sycjiy = — Sv4>v = --—S'KpFp, 



dhy 



the second derivative of the ternary form S\pFp along the normal, 

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

 property of S-yV we thus have 



S(T<l>a = 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, 



,= SX--51^,f, (U2) 



where the factor y- 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 ySrjK, when |8, 

 the axis, is written for p. Therefore 



Ta = TySri\, (114) 



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

 and sufficient that 



(yv^ - S^yV)VpFp 



shall be parallel to 77 when /3 is put for p after the differentiation. 



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

 direction which Vpy takes when /3 is written for p, so that a, jS, and 

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

 length, 



V2 = -S^aV - S2/3V - /S27V 



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

 /3 is an axis. This gives 



^2y2 _ S2^y = +S^aV, 



