S1E B. C. BEODIE ON THE CALCULUS OE CHEMICAL OPEEATIONS. 
'43 
unit of hydrogen is not “ distributed,” and is necessarily (unless we make totally un- 
meaning assumptions) to be expressed by one prime factor a. If, therefore, we are to 
depart from this principle some ground must be shown for so doing. 
The comparative value of two hypotheses may be tested by following these hypotheses 
out to their consequences, and comparing these consequences with the facts to which 
the hypotheses are to be applied. If we find that one hypothesis is applicable to all 
the facts, while the other is applicable to only a portion of those facts, the latter hypo- 
thesis is inadmissible. But both hypotheses may include all the facts, and be equally 
applicable to them, including in each case precisely the same facts. In this case both 
hypotheses are equally tenable, and we have no means of giving a preference to either. 
But there is another case : both hypotheses may include all the facts, and so far either 
may serve our purpose ; but one of the two may also do more than we want, covering 
more ground than is required, and indicating improbable results. In this case that 
hypothesis is to be preferred the consequences of which most exactly coincide with the 
facts, namely, the more restricted hypothesis. 
Assuming, then, that we are in possession of the symbols of the units of ponder- 
able matter constructed, with logical precision, upon these two several hypotheses, 
let us consider the consequences to be deduced from the expressions thus assigned to 
them. 
To simplify the problem, let us take the case of the compounds of hydrogen, chlorine, 
oxygen, carbon. 
If to units of any chemical substance be made up of p units of hydrogen, p' units of 
chlorine, q units of oxygen, and r units of carbon, to, p,p', q, r being positive integers, 
what are the relations, if any, by which these integers are connected 1 We shall assume 
m,p,p\ q, f to have no common measure. 
I. We will first consider hypothesis a\ On this system we have : — 
Symbol of Hydrogen . . 
„ Chlorine . . 
. . a 2 , 
. . c 2 , 
Oxygen . . . 
Carbon 
the last two symbols being identical in the two systems. The symbol, then, of a unit of 
matter, expressed by the prime factors «, c, £, is o n c n '^ n "x, n '" [I. Sec. V.], whence, 
from the conditions of the problem, we have 
to a n c n ' % n " xJ 1 "' =pa? -\-p' c 2 -f q^+rz * ; 
whence [I. Sec. V. (10)] 
nm—2p, 
n'm=2p' : 
MDCCCLXXVII. 
H 
