1863.] 675 



1-f-a, 1+/3, 1+y, . . . 1+X, then in order that the system may 

 have a resultant, since the number of ratios to he eliminated is 

 +/3 + y-f .. . -fX this sum must be equal to n. 

 Let 



and let 



LLpL^ . . .L n =P, then 



1st, the degree of the resultant in question in regard to the coefficients 

 of the rih equation will be the coefficient of f , <f . r y , . . o/ x in _.. 



2nd. As regards weight. By the weight of any letter in respect to 

 any given variable is to be understood the exponent of that variable 

 in the term affected with the coefficient ; and by the weight of any 

 term of the resultant in respect to such variable, the sum of the 

 weights of its several simple factors ; each term in the resultant in 

 respect to any given variable has the same weight ; and this weight 

 may also be proved to be alike for each variable in the same set, and 

 may be taken as the weight of the resultant in respect to such set. 

 This being premised, we have the following theorem : 



The value of the weight of the resultant in respect to any particular 

 set of the variables, ex. gr. the (1 +a) set, will be the coefficient of 

 f i +a . a? . rv . . . a/ in P. 



In the particular case where a=/3=y . . . =X, the above ex- 

 pressions for the degree and weight evidently become polynomial 

 coefficients. Thus, ex. gr., if we suppose each equation linear in 

 respect to the variables of each set, the degree of the resultant in 

 respect to the coefficients of any equation will be 



7rQ+/3+y... 



7T66 . 7T/3 . Try . . . 7T\ 



and its weight in respect to the (1 -fa) set will be 



7r(l+a)7r/3.7ry...7r(\) 



In particular if each set is binary, so that a=/3=y . . . ==X= 1, the 

 degree becomes TT(W), and the weight ^ '. 

 The above theorems are, I believe, altogether new. 



