812 SIE B. C. BEODIE ON THE CALCULUS OF CHEMICAL 0PEEATI0N8. 



•whence 



2?n^2+w and 27Wi=ni, 

 to which is attached the condition 



w(ce)=l, 



m+miw(|)=9; 



1 and 9 being the densities of hydrogen and of water, and w(a) and tv(^) being positive. 

 The integral and positive solutions of these equations as regards m, m^, n, Wj, give all 

 the possible hypotheses which can be made as to the components of oxygen and of water, 

 which are consistent with the hypothesis that the unit of each of these substances is 

 composed of an integral number of simple weights, and that the unit of hydrogen is a 

 simple weight, and the minimum solution selects from these that one hypothesis which 

 is both necessary and sufficient to satisfy the condition given in the equation 



2^=2^1+92- 

 This solution is 



n =0, m =1, 

 111=2, mi = l, 



whence the symbols of water and oxygen as determined from considering the above 



equation are 



Symbol of water a|, 



Symbol of oxygen |^, 

 and the relative weights corresponding to the prime factors a and | are 



the equation being thus expressed, 



2a|=2a+|2. 



It is not to be assumed without proof that these symbols will satisfy the conditions 



afforded by other equations. This is a matter for inquiry. But we have arrived at the 



knowledge that no symbol can be found for these substances composed of a smaller 



number of prime factors, and also that if these symbols can be so expressed the indices 



of these factors will be found among the integral solutions of the above equations, which 



are given in the forms 



m =l + t , n =2t, 



mi=l+^i, ni=:2(l+^i). 

 Hence we arrive at the following general forms for the symbols of oxygen and water, 



Oxygen a='f <!+''', 

 Water a'+'S'^'', 



which include all the possible forms of symbols which satisfy the above conditions. 

 From the equation 



