694 DR ALEXANDER CRUM BROWN ON AN 



\^ being here the replacement of H by C 2 H 5 , 0\^ is the replacement of H by 

 C 2 H 3 , in which H is replaced by CH 3 ; <p 2 on the other hand, is the replace- 

 ment of H by CH 3 in which H is replaced by CH 3 and <£'</> 2 , is the replacement 

 of H by CH 3 in which H is replaced by CH 3 in which H is replaced by CH 3 . 



IV. — On Negative Integral Indices as applied to Operators. 



Without contradicting any previous assumption, we may define the symbol 

 (p~ x by the equation (f>-<p~ l -a = a, or </> - -1 = 1. It is at once obvious that 

 (p'(p~ l — (p~ x '<p, for if cfi be the replacement of A by B, <£ _1 is the replacement 

 of B by A, and </>-<£ _1 is the replacement of B by A, in which A has been 

 replaced by B, that is, the replacement of B by B, and similarly <p~ l '(p is the 

 replacement of A by A ; it is different, however, in the case of vertical multiplica- 



tion, 



•r 1 



fh- 1 



a is not necessarily equal to a, for , 



•a expresses the replacement 



of A by B in one part of the molecule, and of B by A in another part of it, and it is 

 only in particular cases that these two processes will leave the molecule unchanged , 



•a must, of course, be isomeric with a. 



<P 

 <P 



J*_i •« is from the last chapter (cp~ l \ •«, and no confusion will arise by 

 writing this <p~^-a, as the only other meaning which this symbol could have 

 would be (</> 2] ) 'a, and this would be defined by the equation <p' 2] '(<p 2] ) '« 



= a, which could only mean ^.?-ij*«, giving as the equivalent for (</> 2] ) ; the 



expression fcp- 1 ). We have thus, generally, j_„ 



— m] 

 ] 



■a = 0- (m +"W«a, and 



p] 



</>'■ 



a (where p, q, r, &c. are any in- 



/ \"] . (Ml 



( <p~ nQ j -a — cp~'"" ] -a. An expression, T r] 



&c. 



egers, positive or negative) can then always be reduced to the form ^_ fl] -a, by 



adding all the positive indices to form m, and all the negative indices to form n. 

 In the case of horizontal multiplication, we have at once (a-ft-'y . . . . v) _1, « 



= v -1 .... y~' 1 -^~ 1, a~ lm a, and therefore fcp" j = (0 -1 ) ' an( * we mR y^ 

 therefore, write this <p~ n ; and as cp-cp~ 1 -a = a, we have (p 2 -cp~ 2 -a = <p~ 2 ■ cp % • a 

 = a; and, generally, <p m -(p- n -a = cp m -"-a; also (<£- w Va = (p- mn -a and 



