DESCRIPTION OF A NOTATION FoR THE LOCIC OF RELATIVE 



1 



I I 



The differentials of functions involving backward involution are 



(148.) 



(149.) 



d n x 



tl'l 



j«j n — l x,dx . 

 x i,dx \og.x . 



In regard to powers of 6 we have 

 (150.) 



xQ 



6*. 



Exponents with a dot may also be put upon either hide uf 



affect. 



The greater number of functions of x in this algebra may be put in the form 



(pX 



Z p S 9 P A 7 Px'i ; >B 7 . 



For all such functions Taylor's and Maclaurin's theorems bold good in the form. 



(151.) 



dx 







y 





oP«' 



/. 



The symbol 



is used to denote that a is to be substituted for b in what follow* 



For the sake of perspicuity, I will write Maclaurin's theorem at length. 



q> x 



x 

 dx\ 







X 



1 .*>• + irf- + ' 



I 



o! * is 21 *' 



3 



The proof of these theorems is very simple. The (/> + ?) 





differential of 



the only one which does 



Thi< differential th 



lp + q]\ P(dx)<*. It is plain, therefore, that the theorems hold when the coefficient* 

 vk<i and PT& are /. But the general development, by Maclaurii.'s theorem, of a 9 x or 

 ( V x)a is in a form which (112) reduces to identity. It is vety likely thai the apph- 

 cation of these theorems is not confined within the limit* to which I have reacted 

 it. We may write these theorems in the form 



(152.) 



y 



tx 







y 



Qd 



f, 



provided we assume that when the first differential fa positive 



6*/ 



> + r: 



1 d 1 + r?<£ + etc - > 



2\ 



but that when the first differential is negative this becomes by (111), 



Qd 



l+d. 



VOL. IX. 



49 



