APPLICATION OF MATHEMATICS TO CHEMISTRY. 693 



shall assume for the present, that as the operators act independently, their order 

 may be varied without change of meaning, and that the commutative law of 

 multiplication applies.* In the second case, which may be called the horizontal 

 multiplication of operators, it is obvious that the order cannot be changed without 

 changing the meaning, — here then the commutative law does not apply, <p y^-a 

 being in general different from ^-cp-a. 



III. — The Laws of positive Integral Indices, as applied to Operators. 

 In the case of vertical multiplication, we may represent T ya by the symbol 



</r ] -tf, -a, by <£ 3] -#, &c, and generally, it is obvious, that r n] 

 



a = w +"J-a 



and (<p ml ) a = $> m " ] •#, when m and n are positive integers. 



In horizontal multiplication, <p-<p-a, may be written (p 2, a, (p'qy'rfya, (p 3 'a, 

 &c, understanding that ^-(p"-a is merely a contraction for ^-q>-(p • • • • <£•#, 

 and also, that <p n, ^-a is <p- <p • • - - <p ^«; in other words, that the complex 

 operator <p n always acts by means of the <p at the extreme right of the series, and 

 that an operator acting on it always acts upon the (p at the extreme left. With 

 these assumptions, we at once see that the laws of indices cp m -<p n -a — (p m+n -a, 



and ((p m ya = q> mn -a, hold good also in the case of horizontal multiplication 



when m and n are positive integers. 



It also follows from these assumptions, that if $ n = ^, >//•</> = <p n+l , but <p-^ 

 may have several values, one of which is (p n+1 , for in (p'<p'\ the action of cp is 

 restricted to a particular part of <£", while no such restriction is made in the case 

 of <£-\^. Thus, if a be ammonia, and (p the replacement of H by CH 3 , <p 2 -a is 

 ethylamine, and <p'(p 2 a = </> 3, <z is propylamine; but if we put 2 = ^, (p-^-a 

 may be either propylamine {(p z -a) or isopropylamine (^ h <p-a). The graphic 

 formulae will make this more obvious. 



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a (p-a <p 2 'a <p-<p' i -a = <p s a = (p-^'a cp- ] -<p-a = <p^-a 



* It is not by any means certain that this is true, even in the case of the simplest operands ; it 

 is almost certain that it is not true in the case of complex operands ; but as we have not sufficient 

 data to enable us to form a theory connecting the order in which operators are applied to a molecule 

 with the parts of the molecule upon which they act, I have provisionally assumed the simplest 

 possible law. 



