12 



Prof. Sylvester on Binary Qualities. [Jan. 10. 



any irreducible co-variant to the system has for a superior limit 



n 2 — 3 



Thus, eaj. gr., where there is but one quantic, the limit is — — - or 



Z 



-, according as the degree is n odd or v even. 



2 



Secondly, as to order. 



As the expressions become somewhat complicated when there are 

 several quantics, I shall confine myself to a statement applicable to a 

 single quantic, distinguishing between the three cases when n (its 

 degree) is evenly even, oddly even, and odd. 



A. When n contains 4, the superior limits for the order of the inva- 

 riants and covariants respectively are for the former ^ n — ^ ^ - — ^ 



and for the latter 



(n + 2) (n-3) 



B. When n is even, but not divisible by 4, and is greater than 2, 

 the limit, for the two species are and (" + 2 > f^gL 



respectively. 



0. When ?i is any odd number greater than 3, the order of the 

 .3 



invariants has for its limit - (n +1) (n — 3), and when it is any odd 



number greater than unity, the order of the covariants has for its 



3% 2 — 4%— 9 



limit 



2 



Further investigations will, I have good reason to believe, lead to 

 considerably lower limits than those given for cases B and G. 



Although morally certain the three formula? A, B, G cannot be con- 

 sidered at present apodictically established, the formula respecting 

 the limit to degree may, I believe, be regarded as admitting of a com- 

 plete demonstration. There exists, however, a superior limit to the 

 orders of the fundamental invariants or covariants, which may be 

 regarded as subject to direct demonstration even in our present state 

 of knowledge ; this when n is even is n 2 — 2u — 3 for invariants, 

 and n 2 — n — 4 for covariants ; and when n is odd, the corresponding 

 limits are 2n 2 — Sn — 5 for invariants, and 2n 2 —2n — 5 for covariants. 

 But I have no moral doubt whatever of the validity of the formulae B 

 and G as they stand, and next to none of the validity of formula A. 



