( 530) 
we determine in the z-plane the region G,,; + G,",; as the conform 
representation of the w-circle | u | = k or in other circumstances 
of the w-circle |w| —= |a|. Together these regions give after 
subtraction of the region Gy already found the region G,"; the 
remainder of the region G, will be the region Gj. 
It is a fact that according to the nature of the given function 
F(e) the regions G, and Gs can assume forms widely differing, but 
the manner is most simple in which the function exercises its 
influence on the form of the series of polynomials (III). 
As long as g(w) is retained, it is sufficient to replace in this series 
everywhere FM (0) by g®(O), to change the development of (+) 
into the development of another function g(x). So it is to be recom- 
mended if we search for the development of /’(x) to regard before- 
hand the simplest function 1:1—2, which according to (IIL) assumes 
the following form 
x Dee 1 
SI ENE eh NA EAGER 
If we write the terms Z'n (er) of this series in the form 
h=m 
Fn (x) = ines Br, h ah, 
== 
it follows that the first coefficient B,1 is always. equal to 
1 
Er Cm) (0), 
ml 
and so equal to the coefficient of w in the power series into which 
g(u) can be developed. | 
To determine the general coefficient B”/1) we can generally 
1) By Argoaasr was given in his /Caleul des Dérivations (1800)” the development 
of F(a tay+tay?+... .) according to powers of y. If we put a =0, a; =); x 
and if we replace y by w we have the development of F(x (t,u-+ bou +... -)] 
=F (xg(u)) according to ascending powers of w and to determine the coefficients 
we can follow the rule found by ArsoGast and called by Carrer “the rule of the 
last and the last but one”, 
