SIR B. C. BRODIE ON THE CALCULUS OP CHEMICAL OPERATIONS. 813 



we have, substituting for m and m^ the above values, 



w{l){l+t,) = %-t, 

 whence t is not greater than 8. If ^=8, either w(^)=0 and $=1 (Sec. III. (1)), 

 or <j= — 1, when «^=0, and mi=0; in which case «;(!)= - and may have any value. 



In either case we have 



Symbol of oxygen a'^. 

 Symbol of water a^. 



In this case, therefore, owing to the peculiar numerical relation which subsists between 

 the densities, it is possible to express all the symbols by means of the one factor a. 

 The different forms of symbol are given by assigning to t and t^ all possible values ; 

 thus, for example. 



f=0 



^=1 



<=2 



t=Z 



t=l 



#=8 



The problem before us is the selection from this system of symbols of that symbol for 

 oxygen (if such can be found) which shall satisfy the conditions afforded by the other 

 equations of the system, and in which it shall be expressed by the smallest possible 

 number of prime factors. Our hypothesis must be necessary as well as sufficient. 

 Owing to the peculiar and simple laws which prevail in the actual system of chemical 



MDCCCLXVI. 5 s 



