OPERATIVE DIVISION 399 



Some useful inverts and other details will be given in the 

 tables at the end of this paper. 



(3) This is not the place to make a minute algebraic study 

 of the series ; but some verifications and necessary corollaries 

 are now furnished in the form of examples. 



E. Invert (x -+■ i)' 1 = 1 



by descending division. The invert vanishes. 



F. W 4 =[^]">o 



r \ 1 - - 



This identity is obvious since [ [o ,l ][^] J _1 [o' 1 ] = [^ r »]" 1 [ *]L 0W ]' 



That is, if ^ n x =y, then y($> H x)= Vy, as often written. In 

 fact it can be seen at once from the series that 



*„-» = [0 - l -W - {(-) c + (l) ^lo- 1 - . . .][oq. 



n [\n/i \n/2 ) 



Hence we infer that the superior operation on the right side of 

 this equation is nothing but the invert of y(^») ; and if we 

 expand this last operation by the multinomial theorem and 

 then invert it by descending division, the supposition will 

 prove to be true. This enables us to reduce descending division 

 by any such form as yjr n to one of simple descending division by 

 the form yjr x . For example 



[o» + 6o]- 1 = [v/(o» + *o)]-Vo = [o+%° -lb i o~ 1 +. . .]-Vo. 



In this case, however, the form ^1 will contain an infinite 

 number of terms. 



Y T 



G. Find the values of (</>,; x ) and (^h; 1 ) ; 



that is, the rth algebraic power of the inverts. These can be 

 obtained at once by the multinomial theorem ; or by dividing 

 o r instead of o by <f> H and yjr nf since 



1 



Or they can be obtained by dividing o by [</>„]o' T . The results 

 are (" Verb Functions," p. 59), 



{t tf-*+-U-Z±±U-* ! +-Z r \- , -±±\*?+... 



(f j . ; + _L. ( _L=1\ & + -J-i - r -=±\ ^ + . . . 



v T ' r — 1 1 » 1 r — 2 I « J 2 



