692 DR ALEXANDER CRUM BROWN ON AN 



II. — Combination of Symbols. 



The sign + will be used in a purely enumerative sense, to connect operands. 

 Thus, a + b + c + &c, means one molecule of a, and one of b, and one of c, &c. 

 In the same way na 'means n molecules of a, the sign + and numerical multi- 

 pliers never being used to indicate combination. 



Many chemical processes are capable of being applied either to one molecule or 

 to several simultaneously. Thus, if <p be denned by the equation a + NH 3 = (p-a 

 + 2HC1, or by a series of equations equivalent to this, (p may be performed either 

 on one molecule containing two atoms of chlorine, or on two molecules, each con- 

 taining one. In the latter case, a becomes a + b, and (p-a becomes cp-(a + b), 

 here cp is not performed on a and on b, but on (a + b)\ that is, partly on a and partly 

 on b, so that <p-(a + b) is not equal to (p-a + <fib. The distributive law does 

 not, therefore, hold good here, (a + b) is, however, the same as (b + a), and 

 generally we may change the order of operands connected by the sign + . 



In (p' (a + b + c + &c), when a, b, c, &c. are identical, we have cp-na, and 

 just as <p'{a+ b + c + &c.) is not equal to (p-a + (p'b + (p-c + &c, so cp-na is 

 not equal to n (pa; in other Avords, the commutative law of multiplication does 

 not hold good for operators and numerical multipliers. 



If a new process \j/ be performed on the operand <p ■ a, there are two ways in 

 which it may act ; 1st, the process ^ may act independently on a, thus if a — CH 4 

 and (p be the replacement of H by COHO, and ^ the replacement of H by CI, (p 



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and \J/ may act independently, giving chloracetic acid, i i ; or the 



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new operator ^ may act on that part of the molecule which has been intro- 

 duced by the process </>, giving in the instance above the acetate of chlorine, 



w rr. Y N''0"ff) may therefore have two quite distinct meanings, 



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^■{(p-a) 

 and it will be advisable to have two forms of notation for it. I shall represent 



the first by ? 



a, and the second by ^' (pa, the complex operator ^-(p being 



in the case cited above, the replacement of H by COHO, in which the H is replaced 

 by CI, and may, therefore, properly enough be said to be the operator (p, acted on 

 by^. 



In the first case, which may be called the vertical multiplication of operators, I 



