OPERATIVE DIVISION 221 



be denoted by an index placed outside the square brackets. 

 For example : 



[o]» =0 [a + o] n = na + o [bof = b n o [o m ]' 1 = o m " 



a 

 M'(i)=a a [log a?(e) = log, a a 



°\oga 



f— 5— T- -J— [a + bof = a L^|" + b' 



L1+0J i+wo L J 1—0 



[a + bo T = a -\ ; .... — r-j- , — ; — = 



L J a + a + a + for 1 Li + oj 1 — no 



(5) It is always stated in the books that if <£ is any operation 

 then </h/> _1 = </> _1 <£ = <j>° = 1 . But <£° does not equal numerical 

 unity, and, as argued in "Verb Functions" (p. 36), cannot 

 possibly do so. In fact we shall see, if we use the present 

 notation, that when an operation is performed on the same 

 operation reversed (or the converse), then the result is always, 

 not 1, but 0. For example : 



[bo] [60]- 1 = [bo] [Jo] = ; 



[a + 60]- 1 [a + bo] = p-^] [a + bo] = 0. 

 That is to say : 



w m- 1 = ir l ] m = [«■ = 0. 



What then is the meaning of this symbol ? Obviously 

 it has no numerical value, because it merely denotes, so to 

 speak, the empty space which should be occupied by the 

 principal variable in a function. It really expresses what has 

 been called the identical substitution, since [o] x = x. But 

 when we compare the algebraic law that aa' 1 = a = 1 , we 

 gather that [(f)] °or may also properly be called, not the 

 numerical unit, but the operative unit. This becomes numerical 

 unity only when affected by the algebraic power of zero ; and 

 the error referred to seems to have greatly retarded the develop- 

 ment of operative algebra, even when it is applied to the 

 Calculus under the name of the symbolic notation. With this 

 correction, however, operative algebra becomes as exact as 

 numerical algebra or quaternions. 



(6) In extension of this notation, [$] [^] _1 may be called an 



operative ratio and may also be written ■ — a double line being 

 15 



