APPLICATION OF MATHEMATICS TO CHEMISTRY. 695 



(^pA -a - 4>-" ,n -a; or, the ordinary laws of indices hold good in horizontal 

 multiplication where m and n are any integers, positive or negative.* 



V .—Fractional Indices as applied to Operators. 



The symbols <£» and <£"-! maybe defined thus (<p T >) -a = a and (<p*l) a - a. 



When is of the form \p, the first expression will have a real meaning, and so 



will the second when cp is of the form \p ] . In the opposite case, <p n and re -J are 

 purely imaginary operators, and it is only when n of them are multiplied together 

 in the first case horizontally, and in the second vertically, that they acquire 

 a meaning. It may, however, be useful to break up an operator into imagi- 

 nary factors. Thus, if cf> be the replacement of three atoms of hydrogen by one 

 of nitrogen, 0" is the replacement of H by i N, and where the 3 H atoms replaced 

 by N have different positions in a molecule, it may be convenient to express it in 

 this way. Thus, if a be benzol, \J/ the replacement of H by NH 2 , and <p the 



replacement of 3 H by N, we may express diazobenzol by ^ J a a; thus indi- 

 cating that the process <p is performed on •^•a, so that <£* ] acts on -v^, and 

 (p i] on a. 



In the same way, an operator may be broken up into imaginary factors which 

 are not identical; thus, if <p be the replacement of N by 0" and HO, we may put 



a 



(p- a = a -a where a is the replacement of |N by 0", and /3 the replacement of 



p 



JN by HO. 



VI. — On Chemical Groups and Series. 



Those groups which in a former paper {Trans, xxiv. 331) I have called 

 chemical genera, and which are characterised by containing a particular radical, 

 the generic radical, may each be represented by a generic formula $*X, where 

 </> is an operator introducing the generic radical, and X a molecule or group of 



* There is an obvious and important difference between direct and inverse operators, which I 

 may mention here. 



If we represent as direct those operators only which express direct processes which can be per- 

 formed so as to get <f>'aby acting on a, it is plain that if <£ includes the physical conditions of 

 the process, <f>'a can have only one value. It may have a number of conceivable values, and it is 

 the business of the experimental chemist to find out which of these is the real one ; but <f>~ 1 a may 

 have more than one real value, for c£ — x is not restricted to mean one actual process capable of being 

 performed, but any process, such that tfi~ 1 a shall be by the process <£ (performed on tbe part of the 

 molecule, introduced or modified by <£ — J ) reproduce a. Thus, if <fi be the addition of H 2 , and a be 

 alcohol, <j>~ la ma y represent either aldehyde or oxide of ethylene, for both of these give a when 

 treated by the process <£. Examples of this kind might be multiplied to any extent. 



VOL. XXIV. PART IIT. 9 C 



