SIK B. C. BRODIE ON THE CALCULUS OF CHEMICAL OPEKATIONS. 807 



quantities, the indeterminate equations, whence the value in whole numbers of]),p^, q, q^, 

 r, I'y are to be ascertained, become 



mp -\-m'q +m"r =0, 



mp , + w'3'1 -\-m"r^ = 0, 



the two equations containing three indeterminate quantities. 



If in the original equation two symbols were given as determined from other con- 

 siderations, as that 



s, Sj, t, ti being given positive and integral, the indeterminate equations would contain 

 only one unknown quantity, and the problem would be possible in that case alone where 

 the values of ^ andp^ derived from them were positive and integral, and where the con- 

 ditions before referred to and given in the equations connecting w{a) and w{b) were 

 satisfied. 



The course to be pursued in other cases is sufficiently obvious from the above instance. 

 It remains only to state the nature of the problem in its most general form. 



If there be a system of N equations connecting the chemical symbols (p, (Pi, (p.^, ^3, 



of the form 



where m, m!, m", ... are numerical symbols, negative, positive, or 0, putting as before 



(p =a''b'''c''' , 



(p^=a''b^'c''^ , 



(p.^=a' b''(f^ , 



we shall have N sets of indeterminate equations connecting p,q,r,.., and^i, j'j, rj, ... 

 and P2, 2'25 ^2' • • • ^^ ^^^ form 



mp -\-m'q -{■m"r + =0, 



mpi-]-m'qi-{-m"ry-{- =0, 



mp2+m'q2+m"r2-\- =0, 



If a common positive solution in whole numbers of these N sets of equations for 

 P,2^r, , pi, q^, ri, , p^, q^, r^, 



can be found, then the symbols p, (p^, p.^, can be expressed in the given system of 



equations by means of the prime factors a, b, c, ; if such a solution does not exist, 



then the symbols cannot be so expressed ; and the simplest expression of the symbols 



ip, <Pi, (p2 in that system of equations by means of the prime factors a, b, c, is that 



expression in which the indices p, p^, p^ , y, q^, q^, , r, rj, fg, ha^e the 



minimum integral values which satisfy the above N sets of indeterminate equations. 



5r2 



