5 86 SCIENCE PROGRESS 



This is the same thing as splitting the subject, as referred to in 

 Part II, p. 401. Hence if / be the generating iterand and 

 g = «/y, we must have by the preceding subsection 



Any solution of this functional equation in / may be verified 

 in two ways. First we must show that the iterands Ig, Pg, 

 Pg, . . . constantly approximate to the series on the right. 

 Secondly we must show that 



because if rg reaches a given value, II"g must reach the 

 same one. 1 



I will first show that the simple or axial form of iteration, 

 that is, of ± f/M, as described in Section VI (3), will generate 

 all the series given by operative division if the subject and 

 the modifier be properly selected. We shall see that 



x q = r g = [o -f/qg n ~Tg = [WJ- 1 * 



where )x = o is the original equation of the nth. degree (in 

 descending terms and with p Q = 1), which is divided by x n ~ q 

 in order to obtain the setting -ty- q x — m — p q /po, which yields 

 the invert [y^ 9 ]" 1 ^. Note that / is a rational operation, the 

 modifier qg n ~ l being a constant ; and that this modifier contains 

 a power of the base g. 



To save space, I will verify this proposition in a special case 

 — when q = 3, say, and write the setting 



x 3 — bx 2 — ex — m — cx~ x ... — p n x~ n + s = d — m — y = g 3 . 

 Then 

 I = o- I/35-"- 1 . {0" - bo 11 - 1 - co n ~ 2 —{y + m)o K " 3 - . . . -p n } 

 = - I/3S-"- 1 . {(o B - £ 3 o"- 3 ) - &0"- 1 - co 11 ' 2 . . . - wo"- 3 ...-p n }; 



h = g+ l f + ±cg-* + -mg- 2 + '-eg- 3 + . . . - 3 Png' n+1 J 



Pg=ig - i/zg n - x .{{ig) n -g z {ig) n -* -biigf- 1 - c(igy- 2 - . . .-pa 



., + !* + £ + «*),* + 



+ \-m — (n — 2)-bc — (3W— l)—b 3 \g' 2 . . . 



1 Conversely we can find iterands by solving the functional equation. This 

 may be done by the operative division of a given invert by itself, the form of the 

 quotient /being assumed. 



