806 SIR B, C. BRODIE ON THE CALCULUS OF CHEMICAL OPERATIONS. 



symbols by means of prime factors consist of one equation, and, to render the problem 

 determinate, let the equation contain only two undetermined symbols, and be of the form 



Avliere m, m', m" are known, being positive or negative numerical symbols; and let 

 c.^=a"b"', a and b being the symbols of simple weights, and n, n^ given positive and 

 integral numbers. 



Then putting (p=a>'b''', (Py=a''(f'\ 



ma'b''' + m'^'i'' + wi"a"i"' = 0, 

 Avhcnce, from the fundamental equation x-\-y=x^, 



(a''i''')'"(a'i''')'"'(a'"i"')'""=l, 

 and from the property of simple weights before given (Sec. V. (7)) 



rnp -^m'q -\-m"n =0, 

 mjpi-\-m'qi-{-m"ni=0. 



The integral and positive solutions of these equations as regards j), q, py, jj, if such 

 can be found, will give all the possible ways by which the symbols © and (p^ can be 

 expressed by means of prime factors in the above equation, the symbol <p.j_ being of the 

 form given, and the minimum solution in whole numbers of these equations, as regards 

 the same indeterminate quantities, will give the simplest expression of the symbols by 

 means of prime factors, subject to the same condition. 



The number of admissible forms of these symbols is, however, further limited by the 

 requirement that the factors a and b are to be the symbols of real weights. 



Putting W, Wj, Wj as the known absolute weights of the portions of matter of which 



(p, (p[, p2 ^•J^s the symbols, and w{a), w{b) as the unknown absolute weights of the simple 



weights of which a and b are the symbols, we have for the determination of w{a) and 



w{b) the equations 



pw{a)-\-pyW{b) = W, 



which, subject to the equation of condition 



mW+m'Wi+m"W2=0, 



are equivalent to two independent equations. All values, therefore, of_p, jJj, q, q^ are to 

 be rejected which would give a negative value for w{a) or 'w{b) in the above equations. 



If in the original equation the given symbol ^g be expressed by more than two factors, 

 60 that (p.2=a"b"'c'^, the problem is indeterminate unless the absolute weight of one of 

 the simple weights be given; for in this case we should have only two equations to 

 determine the three values w(a), w{b), ii'(c). 



Or again, if no symbol were gi^en, so that (p^=.a''h'', r and rj being indeterminate 



