the Vialytic Method of Elimination. 223 



from which it is obtained. One prevailing principle regulates 

 all the cases treated of in this and the antecedent memoir, viz. 

 that of forming linearly independent systems of augmentatives 

 or connectives, or both, of the given system whose resultant is 

 to be found, of the same degree one with the other, and equal in 

 number (when this admits of being done) to the number of 

 distinct terms in the functions thus formed. The resultant of 

 these functions, treated as linear fimctions of the several combi- 

 nations of powers of the variables in each term, will then be the 

 resultant of the given system clear of all irrelevant factors. If 

 the number of terms to be eliminated exceed the number of the 

 functions, the elimination of course cannot be executed. If the 

 contrary be the case, but the equality is restored by the rejection 

 of a certain number of the equations, the resultant so obtained 

 will vary according to the choice of the equations retained for 

 the purpose of the elimination. The true resultant will not then 

 coincide with any of the resultants so obtained, but will enter 

 as a common factor into them all. 



The following simple arithmetical principles will be found 

 applicable and useful for quotation in the sequel : — 



{a.) The number of terms in a homogeneous function of jo 

 letters of the wth degree is 



m.{m-\-\) . . (m+jo— 1) 

 1.2 ... ^ 



[h.) The number of augmentatives of the (m + w)th degree 

 belonging to a function oi p letters of the mth degree is 



[n + \){n + 2)...{n+p-l) 

 1.2 ... p 



(c.) The number of solutions in integers (excluding zeros) of 

 the equation fli + «2+ • • • -{-(ip=^k m 



(k-\)(k-2)...{k-p+\) 



1.2 ... {p-\)- 

 To begin with the case of binary aggregatives. Let 



^n[x,y) + Gn-i{cc,y)\' + Gn-i'{a^,y)fi''-\-&cc..,. + G«_(t)(a7,y)(9' 



(0 



Kj,(^,2/) + K^_,(^,2/) V -f K^_,'(^,2/)/.* + &c. ... + K^~ (0 (^,y)<9^'^ 



be a system of functions (whose Resultant it is proposed to de- 

 termine) equal in number to the variables x, y, \, fjb . . . 0, and 

 similarly aggregative, i. e. having only the same powers of \, fM, 



