224 Mr. J. J. Sylvester on Extensions of 



&c. entering into them, but of any degrees equal or unequal 

 m, n, . .p. Let the number of the functions be r. Raise each 

 of the given functions by augmentation to the degree s, where 



5={m + 7i+ ... +jy}-(t + ^'+ ... +W)~1, 



the number of augmentatives of the several functions will be 



{s + l)—m 

 (s-\-l)-n 



{s + l)-p, 

 and the total number will therefore be 



r(s + l)-(m + w + &c.+/?), 

 which 



= (r-l)(w + «+ ... ^p)-r(L-^L'+ ... +(0). 



Again, the number of terms to be eliminated will be the sum 

 of the numbers of terms in functions respectively of the 5th, 

 (5— i)th, (5— t')th . . . (s— (*)th) degrees, which are respectively 



s + l—i 

 s+I-l' 



s + l-ii), 



and the number of these partial functions is r— 1. Hence the 

 number of terms to be eliminated is 



(r— l)(w + 7i + &c.+jo-fc + t' + &c.4-W}— 0+*' + &c. + (t)) 

 = (r-l)(m + n + &c.+;?)-r(t + t'-f ... -f (t)), 



which is exactly equal to the number of the augmentative func- 

 tions. Hence the Resultant* of the given functions can be found 

 dialytically by linear elimination, and the exponent of its dimen- 

 sions in respect to the coefficients of the given functions will be 

 the number 



(r — l)2wi— rX*, 

 as above found. 



The method above given may be replaced by another more 

 compendious, and analogous to that known by the name of 

 Bezout's abridged method for ordinary functions of two letters. 

 As the method is precisely the same whatever the number of the 



♦ The Resultant of a system of functions means in general the same thing 

 as the left-hand side of the final equation (clear of extraneous factors) re- 

 sulting from the elimination of the variables between the equations formed 

 by equating the said functions severally to zero. 



