1903.] Maximum Order of Irreducible Covariants. 399 



" The Maximum Order of an Irreducible Covariant of a System 

 of Binary Forms." By A. YOUNG, M.A., Clare College, 

 Cambridge. Communicated by Major P. A. MAcMAHON, D.Sc., 

 F.R.S. Keceived September 26, 1903. 



It has been suggested to me that an incidental result of a paper I 

 have recently communicated to the London Mathematical Society may 

 be of interest. In the paper in question it is proved that all covariants 

 of a system of binary ?i-ics are linearly expressible in terms of _ 



(i) Covariants of the form 



here 

 (ii) Covariants having a symbolical factor 



2 s - 2 , A. < 2-3, 



(iii) Products of covariants. 



Mr. J. H. Grace, in a note appended to the paper, has deduced from 

 this result a means of calculating the maximum order, in the variables, 

 of an irreducible covariant of a system of quantics. If no quantic of 

 the system is of order exceeding n, the maximum order of an irre- 

 ducible covariant is the greatest of the numbers 



where 8 is an integer, in fact, the degree in the coefficients of the 

 covariant in question. 



If n = 2* + k where k ^> 2'', it will be seen that the maximum is 



The covariant of maximum order is then 



There are strong reasons for believing this covariant to be actually 

 irreducible ; in the contrary case a reduction is obtained for certain 

 forms classed as perpetuants.* 



* MacMahon, ' Camb. Phil. Trans.,' vol. 19, p. 234; Grace, ' Lond. Math. 803. 

 Proc.,' vol. 35, p. 107. 



