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, by 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 



V<Pp6p + pS8p, (197) 



where (}> and d are linear in p, and 5 is a constant vector.^ The first 

 term of this vector, the vector product V4>pdp, has three zeros, for 

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

 the same manner by the operations <^ and 6. These directions ^^ are 

 the axes of the linear vector function (i>~^d. Let three vectors along 

 these directions be jSi, ^2, and /Ss. Let them be converted by 4> into 

 Xi, X2, Xa, respectively. We may then write 



#•(123) = Xi(23p) + X2(31p) + X3(12p), 



or with the notation already adopted, 



4>p = XiOTi + X2a-2 + X3.T3 (198) 



and also 



Op = gih\X\ + g'^2X2 + gz>^zxz, (199) 



where gi, g^, and gs are the roots of the cubic in ^~^0, that is they 

 satisfy three relations of the form </)-^ 0/3 = g^. Whence it follows 

 that 0/3 = g(l)^ and F0/30/3 = 0., for 181, /S2, 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. Joly to Hamilton's 'Elements of 

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



