264 THE REV. F. H. JACKSON 



in which 



a = / r«i"+ r _ r«ir«. - 1 1«+* 4. . -i. i _ 1 \»«w»-«_ 



«-«]![*]! 



A, = | [«]"+'' _[„][« -!]»+>•+ + (-l)«gWM)JW [n-s] n+r + 



,(_ lW »- 1 »(»- a » W }_il=^_ )j . (25 ) 



+ i 



provided that the functions, on which the operations are performed, are capable of 

 expansion in Maclaurin's form. The coefficient A r is the g'-equivalent of the number 

 A".0" ( tabulated by De Morgan and Boole (Boole's Finite Differences, pp. 19, 20). 

 The development of such expressions as A"$(a;)^a; may be shown to be dependent on 



A<>(.r)<%))={A^ + QA;}{A,«- 1 + QA>} .... {A. + QAV}!^)*^)} 

 in which 



A q r4>(x) = <t>(qx)-q r 3>(x),A^ = \,,\n +1 . . . A^A, , Q and A 



operate on &(x) only. I leave the consideration of these forms, however, to another 

 paper. 



§ 8. Forms of Maclaurin's Series. 

 In (24), putting 



A M *(a:) : 



we see that 



M0+MJ 



d\ n+1 

 dx) 



*(*), 



i A"*(a:) 1 = A $<"»(0) , 



= { W " MO -lf+ + ( - m^^^f.n - sf +...(- l)y»-*->] } J^,, 



= K ! $N(0), 



so that Maclaurin's series takes the following form in g-function theory : 



say *( a O = *(0) + ^A*(0) + ^ ! A**(0)+ 



and a conjugate form 



<D(a;) = $(0) + -^4.(A.0)0 + ^ ! <E>(A.0).0 2 + 



where <I>(A.0)0'" denotes what <I>(A)ce'" becomes when x = Q. 



The special development of this and the analogue of Taylor's theorem I defer to 

 another paper. 



