44 ^ Mr. HEttsdHEL on various points of Analysis* 
deVelopement of f ( A^), be every where written for { | 
and to derive the generating function of vA^ from that of A^, we 
have only to multiply the latter by / (y), thus 
^ ^ -/(t) ; • • • ( 2 ) 
Let now v'^A , v^A , &c. denote the respective values of the 
expression 
^A 
+ 
X-\- a, * JT -j- ^ 
When instead of A^ we write vA^, V"*A^, &c. that is, 
v'"^' + &c. 
and it is easy to see that we shall have 
G = {/(t) }"•'?’ (0; . • • . (3)- 
Let us now denote by / |yj the expression 
•a . r" + 'b . + &c. 
and by V*A^ the function formed from f |yj in the same 
manner as we formed v Ax from/ |yj. The equation ( 3 ) 
then may be thus written 
G{v‘AJ = {(/ r ■): y ( i )}^^(0 
If then {f/”"* (^)}^ be developed by the ordinary methods 
into a series of the form 
a t 
“ + a/ + . . . . 
we shall obtain 
G { v'A,} =S, { :/(A) {t) = s4«*.g{ V“A«| }, 
