332 HITCHCOCK. 



Fp = V<ppdp + pSbp (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 VippOp may in general be expanded as 



Vippdp = a1.r2.T3 + a2.r3.r1 + a3a-i.r2 (2) 



where ai, a2, and 03 are constant vectors, and where the .r's are given by 



a-iS/3i/32/33 = S/32/33P, o^-^S^^z = ^^:^iP, x^S^m^ = Sl3,^2P (3) 



and where the a's depend on nine scalars by the scheme 



ai = Jiii3i + .l2i/32 + J31/33 



a2 = ^i2i3i + Aool^o + A32I33 (4) 



a3 = ^13i3l + /I23/32 + As3^3 



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

 axes are given, aside from the term pS8p 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 ^'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 A's in terms of the axes j8i, 

 jS2, • • -/Sr 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. 



V^pOp = ^.. ^'''^ ^^-^-^^^ + /3.. ^'^'^ ^^-^-^^^ 

 (P5P6P7) (P6P4P7) 



(457) (P.P.PP) 

 (P4P5P7) ^ 



Now it is evident from (2) that if we let .ri= 0, .T2= .r3= 1, we shall 

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

 Pp to i. We thus obtain 



^_„ (567) (P,Pii) , , (647) (P.Pd) , , (457) (P.Pii) ,.-, 

 "'-'''• (ftP.P,) +'''■ iP,P,P,) +"'• (P,P.P,) (") 



