59 6 SCIENCE PROGRESS 



ascertained. For proximate convergence the terms should diminish quickly — 

 which will generally occur when y is large or when its major coefficients become 

 small. To study the latter case put b = n\, c = n 2 , d=n h . . . ; then all the major 

 coefficients vanish if n is the degree of the equation, since x — y — i. Or we may 

 set the original equation and its invert in what may be called Newtonian 

 coefficients (Table III). 



The calculation of roots is much facilitated by centring the original equation 

 by removing the second term bx n ~ l — which abolishes all the terms containing b in 

 the invert (see Table IV and the latter part of Table II, i). By Part II, 

 Example K, p. 401, linear transformations do not affect the value of the major 

 coefficients of y, which are invariants. 



If y\r n x has no negative powers of x, so that n is the degree of the function, the 

 major coefficients of its invert based upon its absolute term are all functions of 

 the invariants H, G, I, J, etc., as can easily be seen from Table III, for example. 

 These functions may be most readily obtained by inverting the centred form of 

 yfr n x — as, for instance, by putting n.,C, n 3 D, . . . for c, d, . . . in Table IV. The 

 study must be left to the reader, together with that of the serial inversion of 

 quantics in general. 



It will be seen from the formula just given for the general minor term that the 

 latter part of it may be written y(pqly q ) a . (prly r ) b • (psly s Y- • • • Hence the series 

 divided by y consists of all possible combinations of these ratios. Or we may also 

 distribute n amongst them in the same manner, and then rearrange the series in 

 terms of the type pqlnyi collected according to their order. This is done in 

 Table V. The terms may easily be written out to any extent. The major terms 

 now consist of the quantities B, C, D, . . . with the same powers and coefficients 

 as they possess in (A + B + C . . .) e and with the additional coefficients (n — w+i) 

 (2n-w+i) . . . containing e— 1 factors. HA, B, C, . . . are all numerically < 1, 

 the terms will generally diminish rapidly. 



If we rearrange this series in descending terms of n we shall find that the 

 coefficients of the successive powers of n consist of infinite series of the ratios b/y, 

 c/y 2 , . . . which can easily be summed, giving the important logarithmic form 



y- 1 x = y- l \lr~ 1 y n =i - i/nAogG+il(2ln 2 ).-j-(y\og 2 G)- ij(3ln 3 .\ -^-(yo)Jlog 3 L7 + etc., 



where G= 1 + bjy + c/y 2 + . . . = >\r n yjy n . The symbolical 1 sum of this is 



x = yj,~ V =y{ [c " K(? >]°]log G \dy. 



The differentiations here indicated give finite expressions in powers of \ogG, which 

 may be arranged in ascending order, or re-arranged in powers of exp(\ogG/n), 

 and in other ways. All these arrangements are generated by different iterands, 

 such as those suggested on p. 579 ; but I have no space to deal with them here. 



EXAMPLES 

 I have space only for a few examples to illustrate some of the propositions in 

 this article. Note VI and the Tables will facilitate calculations, especially with 

 the aid of Barlow's Tables. The equations are classified according to Note II. 

 O. One-change Equations, A - B = o. 



(1) x 2 -x- 1 =0. 

 By Section VII (3), p. 581, both the positive critical points = 1 ; and both refer 

 to the only positive root and are prospective, i.e. less than it. Which should we 



1 Correct symbolism, however, requires a special Iteration-Operator. 



