664 



NATURE 



[October 29, 1908 



f.,{x) = y and /,(.v) = y. If we have taken it on the wrong 

 curve they will diverge from the required intersection, as 

 will be apparent from Figs, i and 2. The rule to ensure 

 ultimate convergency is that at or near the point of intcr- 

 i ction X, shall be taken upon the curve which has the 

 .umcrically lesser differential coeflScient. If, at the point 

 of intersection, the differential coefficients are numerically 

 exactly equal, the method fails, as x„+„ = j.-„; but the inter- 

 section of the curves will then be at the intersection of 

 the tangents, so that .i. = 5(.\-„^, + a'„) — Fig. 3. It often 

 happens, if we have tal<en .v, at random, that the succeed- 

 ing terms of the series are at first irregular, but after- 

 wards converge. 



If at any stage in the analytical process a term becomes 

 unreal, this means that the corresponding line drawn from 

 one of the curves cannot intersect any branch of the other 

 curve. We must then start again with another value of x,. 



The successive terms may appear to converge for a time 

 and may then diverge. This indicates the position of a 

 pair of imaginary roots (Fig. 4). Compare, for instance, 

 .v"=I3a: — 42, of which the roots are 6 and 7, with 

 .\'"= 13X— 43, of which the roots are imaginary. 



Convergency is slowest when the differential coefficients 

 of the curves at the required intersection are nearly equal, 

 numerically, to each other, especially if both are also 

 nearly equal to +1. It is quickest when their numerical 

 difference is greatest. When we have arrived near enough 



ever, this can be avoided by finding special roots from 

 other forms of the equation. For example, 



x'-3x--2x+5 = o 

 has three roots, 3-128, 1-202, and —1-330, but they can 

 be calculated more easily from the form (x — i)' = ^x—6. 

 One or two of the roots of a complete rational integral 

 equation may frequently be obtained almost at once by 

 dividing the equation by x ~', and putting it in the form 

 x = a+b/x + CiX-. . . . Generally, the first terms of the 

 series x^, x., x, . . . may be estimated mentally, exact- 

 ness being unnecessary until we approach near to a root. 



The rule, therefore, has the advantage of being very 

 L'asily remembered, of giving, theoretically, all the roots 

 in succession, and of leading, almost automatically, to at 

 least one or two solutions. Hitherto it has been considered 

 from the geometric and arithmetric side ; I will now try 

 to indicate briefly its operative and algebraic forms. 



We have evidently to do with repeated operation, which 

 is best e.xpressed by the algorithm of " verb functions " as 

 described in my paper referred to. This algorithm is 

 based on the fact that <j>°, where the inde.x refers to 

 operative and not algebraic involution, cannot possibly be 

 the equivalent of numerical unity (as generally held), but 

 is equal to operative unity. I denote this by the symbol 

 8 (for base). When substituted for the argument in any 

 expression, $ converts that expression into one which 



to the destination, further work may often be abbreviated 

 by assuming that the intersection of the curves nearly 

 coincides with that of the tangents. Thus if -v, and x, 

 are successive convergents on the curves f,ix) = y and 

 fjx) = y respectively, then, approximately, 



fx(x,)-f^{x„) 



In the case of the equation .\'' = i5.v-)-4 the roots were 

 obtained successively by taking x^ alternately, first on one 

 curve, i5A.-|-4 = y, and then on the other, x^ = y. This can 

 be done very frequently, but sometimes one of the curves 

 makes such a bend between two intersections that, by the 

 rule already given to ensure convergency, -r, may have to 

 be taken on the same curve for two roots in succession. 

 By plotting the curves roughly on paper it is generally 

 easy to see at a glance how best to commence and conduct 

 the process (Fig. 5). 



.^s is evident from the geometrical interpretation, the 

 method is by no means restricted to rational integral equa- 

 tions. The two curves /2(.v) = y and i,(x) = y may be any 

 we please, provided only that we can obtain .v = /I-'(y), or 

 at least can evaluate it for different values of y, and it is 

 generally easy to put the original equation /(,\-) = o in such 

 a form that this can be done ; but in order to find some 

 of the roots it may be necessary to take .t, on the other 

 side of the equation, which requires us also to obtain .v 

 from -v = /,-'(y), which may be difficult. Generally, how- 



XO. 2035, VOL. 78] 



denotes an action, not a substantive. Thus a + b&-\-cfi- is 

 the action performed on x in order to convert it into 

 a + hx-\-cx-. 



Now let .r"-l-rtA"~' l-Za""' . . . =/-, the number of terms 

 being unlimited. Then 



.v = [s//--«B"-i-^./8"-" . . .].v, 



where the square brackets denote that the expression con- 

 tained within them operates on the following matter, and 

 is not multiplied into it. Thus x on the left side of the 

 equation is the result of an operation performed on itself. 

 Similarly, x on the right side of the equation is the result 

 of the same operation performed on itself as many times 

 as we please. Hence we obtain the identity 



where q denotes operative involution and may be any 

 integer, positive or negative, to infinity. Now the ex'- 

 pression on the right can be developed with the aid of the 

 multinomial theorem by successive substitution, accord- 

 ing to the common algebra of verb functions, and we 

 obtain 



.v = /t"- 



iTi 



/'+ i 



\n} 2!/ 





t\ ab'r 



+ terms containing x. 



