432 HITCHCOCK. 



PART FOUR. 



33. Our analysis of quadratic vectors is now complete, in the 

 sense that the nature of any given vector can be completely deter- 

 mined by the foregoing methods. Furthermore, normal forms, or 

 model vectors, have been given including all possible vectors. I 

 shall apply what precedes to the study of a general, and very simple, 

 normal form, b}'' means of which the properties of a quadratic vector 

 are made to depend on those of a pair of vectors of the first degree; 

 namely 



V6pdp + pS8p, (197) 



where 4> and 6 are linear in p, and 5 is a constant vector. -"^^ The first 

 term of this vector, the vector product V<t)pdp, has three zeros, for 

 there exist three directions, in general distinct, which are altered in 

 the same manner by the operations 4> and 6. These directions ^° are 

 the axes of the linear vector function </)~^0. Let three vectors along 

 these directions be /Si, (32, and 183. Let them be converted by (j> into 

 Xi, X2, X^, respectively. We may then write 



<^p.(123) = Xi(23p) + X2(31p) + X3(12p), 



or with the notation already adopted, 



</>p = Xia:i + X2a;2 + Xs-Ta '(198) 



and also 



ep = fiTiXiXi + 5'2X2a:2 + .93X3X3, (199) 



where gi, gi, and g^ are the roots of the cubic in 4>~^6, that is they 

 satisfy three relations of the form <^~^ 0/3 = gl3. Whence it follows 

 that d^ = g4>^ and V4>^B^ = 0., for /3i, 182, or ^3. If we now multiply 

 together the corresponding members of (198) and (199), introducing 

 the notation ■ 



19 In a former paper, (Phil. Mag. Jan. 1909, page 124) I gave this form 

 (without proof), in connection with differential operators of the second order, 

 the symbol V being written for p) . 



20 Cf. the appendix by the late Prof. C. J. Jolj^ to Hamilton's 'Elements of 

 Quaternions,' 2nd. Ed. p. 363. 



