Roche — The Quadratic Vector Function. 93 



functions, and no generality is lost if ^i, ^o, ^3 are taken self-conjugate. So 

 the general quadratic vector function of p is of the form 

 (iiSp<j)ip + OiSpipip + a^Spipsp, 



where ^1, ^2, <p% are self-conjugate linear functions. On account of this 

 occurrence of three functions, some results on the invariants and associated 

 functions of three linear vector functions have been prefixed by way of 

 introduction. The symbolic cubic of $ is found, where 



<l* = /^l (^1 -1- A^2^2 + n'3^3, 



and on working it out in the ordinary way the results of § 1 are evident. 

 It is not difficult to prove that the scalar functions an, ha, and ^123 are 

 invariants. The methods in the introductory paragraphs are developed from 

 an unpublished MS. of Professor A. W. Conway. They are perfectly general, 

 and their extension to any number of functions is easy. All the relations 

 between the invariants and concomitants of the three functions have not 

 been worked out. In particular, in the case of the results in §§ 5, 6, only 

 those have been written down that were wanted in the actual working. 



The invariants and concomitants have not now the same meaning as in 

 the case of the linear function. Defining them as in § 12, the subsequent 

 working is not difficult ; yet the results are interesting. Then an attempt 

 has been made to classify these functions by examining conditions that make 

 them binomial or monomial, and not trinomial. The results are not satis- 

 factory, and the general investigation is very difficult. Some particular cases 

 are easy. 



The very important question of inversion has only been touched. It is 

 not satisfactory either for the same reason as above. One particular case can 

 be fully investigated. Before the general question of inversion, something 

 has been said on the number of roots for which the function vanishes. Of 

 course this might be regarded as a particular case of the problem of inversion. 

 A word on what Professor A. W. Conway calls the " central axes " of the 

 function concludes the notes. 



I. — Invariants and Covariants of Three Linear Vector Functions. 



1. If 



$ = ki^i + k,(p2 + k3<l>3, 



<P satisfies a symbolic cubic 



<f ^ - 77i"^^ + m'tp - W! = 0, 

 where 



(Ai) m = Jcihrii + \^7rh ^ h^rris + A-ffeaij + k^'kiani + ki^^a.^ 



+ kz'ham + k^^kici^i + ki'-haiz + duskihks, 

 [15*] 



