406 



HITCHCOCK. 



because VTa = (f)'v, -when da = <j)dp.^^ The numerator of (104) may 

 be written in a number of remarkable forms. It is, for example, 

 the Hessian of the ternary form S\pFp, multiplied by a factor inde- 

 pendent of Fp. This follows from the fact that H is Hamilton's 

 m-in variant for the function ^; that is, if i, j, k, are ANY three 

 diplanar vectors, 



S4ji(j)i4>k 



H = 



Sijk 



(105) 



Choose, as three convenient vectors, p, a, and Vpa. Then 



H 



S(f)p(})(Tcf)VpC 



SpaVpa 



(106) 



By a well-known expansion ^^ we have 



<t)Vp(T = — SV(T- Vpa — V(f)pa — Vp4>(j 



(107) 



If we write this value for (j}Vpa, and multiply out, remembering that 

 </)p is parallel to o- because we deal with homogeneous functions, while 

 Sp<x vanishes on the cone, we have H equal to the numerator of (104) 

 aside from a factor which is a constant multiple of p^. 



More important for our present purpose than this connection with 

 the Hessian, is the fact that the numerator of (104) can be obtained 

 by differential operations performed directly upon the vector VpFp. 

 For, taking the first term of this numerator. 



SVa = V^S\pFp = SW-pFp, 



(108) 



because V^ is a scalar and commutative. As to the second terra of 

 the same numerator. 



Sactxx = — (j^Sv4>v, identically. 



(109) 



If we let 7 be the direction of v at the point on the cone, we have 



15 Proof: dTV = - do-^ = - 2S<Tdcr = - 2Sa<t>dp. Hence v^V = 20V = 

 2TaVTa, and, dividing by 2Ta; we have vTa = (j>'U(t = <^'v. 



16 Hamilton, Elements of Quaternions, Art. 350. Hamilton's m" is the same 

 as —SVcr, and <j> is self-conjugate. 



