ESSAYS 295 



[<f> - O]" 1 ^) ; see especially my papers (4, 5, 6). The geometric interpretation of 

 the theorem of 5*3 can easily be given by such graphs, and the question of con- 

 vergence frequently decided. 



8'0. Expansions of <£" by Maclaurin's Theorem are generally intractable owing 

 to the cumbersome expressions of the successive tangentials ; but the expansion of 

 <£"=<£"{*« + (0 — #z)} by Taylor's Theorem is rendered easy by the simplification 

 due to the definition m=>(}>m, so that also <£*w, cf> 3 m, . . . t^m — m. Hence 



[Zty"]w = [D{<j><p- l )]m = [</>'<£» ~ l . Dcp - *];/* (where Dm d/do) 



-[^.'(jf)"- 1 . <£'<£"~ 2 • &<}>"-' tf>'<f>. <f>']m 



=tym)»; 8-1 



since a subject is distributive throughout an algebraic product of operators. Next, 

 taking logarithms of /?<£" = <£'<£"" 1 . <£'<£"~ 3 • etc., and differentiating again, 



D v= D P\ ^.-1 + ^.-. +• • ■ + -^rr z? * p -i?^ ri D + +etc j 



=<£V _1 • W"- 1 )* + <£'</>"- * . <t>"<t> n -* . {£><!?-')*+ ftp- 1 . <£'<£"-■ . ff-« . 



(£><£"-»)* + etc. 

 Thus[Z>^"]w = ^"w.(^'w)*«- 2 . {i+(^'/«)- 1 + (^';«)- J +.. . + (<jf>'w)-» + 1 } 



, , ^ , ((h'm) n —i 



=*"'"•(*''")"- -^rrr- 8 ' 2 



Hence $*=($> n {m + {O -m)} =m + ($ m) n . (0-w) + etc -> as ' n 6'4- 8*3 



9'0. It is a very familiar proposition that if <$> n x approaches a limit when n is 

 indefinitely increased that limit is a root of (fyx=x, in other words, a midaxial root 

 of (f)X. But how is it that <f> n , which should be an operation, becomes a number? 

 The fundamental rules of this approximation are obvious from graphs, and are : 



9*i. (fi n x ultimately converges to m if | <fim | <i. If <£'/»>o, the iterants 

 <ft a x, <f>*x . . . progressively increase or decrease. If <f)'m<o, they are alternately 

 greater or less than m. Otherwise they recede from m, "stagnate" round it, or 

 diverge altogether. 



9"2. If <f)X > x, the iterants tend to converge to a midaxial root greater than x, 

 or to increase without limit. If (j>x < x, they tend to converge to a midaxial root 

 less than x, or to decrease without limit. 



9*3. If the conditions hold good, the iteration will converge towards tn what- 

 ever value, within certain ranges, the base of the iteration, x, may have. 



9*4. If <f) is a continuous one-branched curve with a series of midaxial roots 

 ...««_ 9 , ot_i, mo, m\, mi, . . ., numbered in ascending order of magnitude and m 9 

 being the least positive root, then <£ B .r can converge only to the even roots m_a, 

 w*o, «a, ... if <£(o) > o, and to the odd roots if <£(o) < o. In both these cases 

 obviously <$m < 1 at each of the appropriate roots, but also by 9*1, we must have 

 <f>'m > - 1 for convergency. 



9*5. Which particular midaxial root, if any, is approached depends upon the 

 figure of the curve and the position of the base x ; but generally <fi n x approaches 

 m r if m r -i< x< m r+ i. 



By 81, [D<f> 1t ]m = (D(f>m) n . Hence if | D$m | <i as required by 9*1 for suc- 

 cessful convergence, [Dcf) n ]m vanishes when n is indefinitely increased ; and the 

 •ame thing happens to the succeeding tangentials. Thus in 6*3 and 8"3, all the 

 terms in the expansion of <£".r except the first ultimately disappear, leaving only 

 <fr*x=m. Or, to put it slightly differently, all the terms except the first in the 

 expansion of <f} H (m+x) vanish, so that this function becomes in the proximity of m 



