332 HITCHCOCK. 



Fp = V^pdp + pSdp (1) 



where (p and 6 are linear vector functions and 5 is a vector. Exception 

 can only occur when one of the axes is of order at least five. 



(C) The term Vifpdp may in general be expanded as 



V(pp6p = a1.r2.r3 + ao.rsri + a3a:iX2 (2) 



where ai, 02, and 03 are constant vectors, and where the .r's are given by 



xiS^i^2^s = S^2/33P, .T2S^i|32/33 = S^^jSip, .r3Si3i^2/33 = S^i/32P (3) 



and where the as depend on nine scalars by the scheme 



ai = .JiiiSi + .421/32 + A:n^3 



as = ^12181 + A.20A + ^1 32/33 (4) 



as = Ais^i + ^23182 + Az303 



(D) In general the quadratic vector is fully determined when its 

 axes are given, aside from the term pSdp and a multiplicative scalar. 



I shall refer to this former paper as C. Q. V. 



2. The A's as Functions of the Axes. 



It follows from (C) and (D) that the nine A's are determinate as 

 functions of the axes, aside from a common scalar multiplier. A 

 knowledge of these functions facilitates our attack on a variety of 

 problems. I propose to express these ^'s in terms of the axes |8i, 

 ^2,* • 'iSy and to illustrate the utility of the results by some applications. 



With the notation and results of pp. 377-384 of C. Q. V. we may 

 write, omitting mere constant multipliers. 



K^pSp = ff,. (567) (P.PM ^ ^ (647) (P,PM 

 (PJ'J'd (P,P,Pd 



,„ (457) (f4P,Pp) ,., 

 + ''•■ (P.P.P,) ® 



Now it is evident from (2) that if we let a-i= 0, .T2= X3= 1, we shall 

 reduce V(ppdp to ai; but by p. 381 of C. Q. V. we shall then reduce 

 Pp to i. We thus obtain 



(567) (P5P6O , . (647) {P,Pd) , , (457) (P4P5O ..^ 

 '''~^' {P.P.P.) +^^' ^P.P.P,) +^^- {P.P.P.) ^^^ 



