40 Prof. Sylvester 07t Derivation of Coexistence. 



This proposition admits of a great generalization*, but we have 

 now all that is requisite for enabling us to arrive at a proposi- 

 tion exhibiting under one coup d'ceil every combination and 

 every effect of every combination that can possibly be made 

 vi'ith any number of coexisting equations of the first degree, 

 containing any number of 7'epeated, or to use the ordinary, 

 language of analysts, (variable or) unknown quantities. 



Art. (9.) For the sake of symmetry I make every equation 

 homogeneous ; so that to eliminate n repeated terms, no more 

 than n equations will be required. 



In like manner the problem of determining ?^ quantities 

 from n equations will be here represented by the case in which 

 we have to determine the ratios of {n -{- 1) quantities from 7i 

 equations. 



Art. (9.) Statement of the Equations of Coexistence. 



Let there be any number of bases {a be ... I), and as many 

 repeated terms {x 7/ z ... ^), and let the number of equations 

 be any whatever, say (w). The system may be represented 

 by the t7/pe equation 



a,x + b.y + c.z+. . .+Z.^ = 0. 



In which r can take up all integer values from — x to -}- oo . 

 The specific number of equations given will be represented by 

 making the arguments of each base ^moc?/c, so that 



a = a , h =■ b , c =: c . . . .1=1 ., 



r (i.n-\-r r (tn-\-r r fiu+r r //.n+r^ 



[ji being any integer whatever. 



Art. (10.) Co)nbi?iation of the given Eqimtions. — 



Leading Theorem. 



Take/, g, ... k Q.s\hQ arbitrary hases of new and absolutely 



independent but periodic arguments, having the same index 



of periodicity {n) as a b c ...I, and being in number {n — 1), 



i. e. one fewer than there are units in that index. 



The number o^ differiiig arbitrary constants thus mayiufac- 

 iured is n . {n — \). 



Let A^ + Bj/ +02+ . . . + L if = be the general 

 prime derivative from the given equations, then we may make 

 A= KYTi{pafg...k) 

 B= rPD(o5/^...A:) 



i = ivW{pifg...h). 



Art. (11.) Cor. (1.) Inferences from the Leading Theorem. 

 Let the number of equations, or, which is the same thing, 

 * See the Postscript to this paper for one specimen. 



