Prof. Sylvester on Derivation of Coexistence. 41 



the index of periodicity (?z), be the same as the number of re- 

 peated terms [x y z ... t), then one relation exists between the 

 coefficients: this is found by malting the [n — 1) new bases 

 coincide with (ii — \) out of the old bases. We get accord- 

 ingly, as the result of elimination, 



^VJy {oahc .,.1) = 0. 

 Art. (11.) Cor. (2.) Let the number of equations be one 

 more than that of the given bases, there will then be two equa- 

 tions of condition. These are represented by preserving one 

 new arbitrary base, as A. The result of elimination being in 

 this case 



^PD {oahc.lx) = 0. 



Ex. The result of eliminating between 



«1 . .2' -1- 5j . j/ = 



ac^.x ■{■hc,.y = 



«3 . .r + ^3 . w = 



is ^ PD (o a b\) = 



i. e. Ag . h^ «j — A3 . 5^ «2 -^ \ . b^ a<^ — • Aj . ^2 «3 -I- ^.^ • ^1 «3 



— A2 .h^a^ — 0, 

 from which we infer, seeing that Ag Ag A^ are independent, 



^2 . a^ — 5j . «2 = 

 5g . «2 - ^2 • «3 = 



§1 . «3 — &3 . «i = 0, 



any fwo of which imply the third. 



In like manner, in general, if the number of equations 

 exceed in any manner the number of bases or repeated 

 terms, the rule is to introduce so many ne*w and arbitrary 

 bases as together with the old bases shall make up the num- 

 ber of equations, and then equate the zeta-ic product of the 

 diiferences of zero, the old bases and the new bases, to nothing. 



Art. (12.) Cor. (3). Let the number of equations be one 

 fewer than the number (?z) of bases or repeated terms; the 

 number of introduced bases in the general theorem is here 

 {n — 2). Make these {71 — 2) bases equal severally to the 

 bases which in the type equation are affixed to 2, ii,..t, 

 then C = 



D = 



L = 0, 

 and we have left simply 



^VT>{oacd..,Jcl)x + ^V'D{obcd,..kl)y^O. 

 In like manner we may make to vanish all but A and C, and 

 thus get 



^VJi{oabd,.,JcV)x-[- ^VD{ocbd..,kl)z = 0, 



