396 SCIENCE PROGRESS 



D. x % — yx + 7 = 



The setting x° -f yx' 1 = 7 appears to converge towards the 

 greatest and the least roots when the positive or negative value 

 of \/j is taken, but in the former case the convergence is 

 very slow. The middle root has been laboriously studied by 



ascending division from the setting x x % = 1 (Section III 



(5) Example D) ; but we may also approach it by descending 



division by putting x = -, which gives z 3 — 7s; 2 + 49 = o. 



As suggested in these examples, any equation can be put 

 into many different settings by dividing it algebraically by 

 different powers of x. This process renders each coefficient in 

 turn independent of x — so that each coefficient in turn may be 

 made the subject of a different inversion, either by descending 

 or by ascending division. We may also obtain a second series 



©f settings by putting x = '■■ and simplifying ; and it would 



appear that different settings yield different roots, while others 

 give quite divergent or inconclusive series, and in others again 

 the subject involves the square root of a negative quantity. 

 Generally speaking, the subject should be as large as possible 

 for descending division and as small as possible for ascending 

 division ; and where it involves a real square root the inversion 

 often yields simultaneously two real roots of the original 

 equation. The reader will probably find considerable difficulty 

 in explaining these things, but may read further before 

 attempting to do so. 



(2) The expansions of fractional or negative powers of the 

 divisor needed for descending division are considerably more 

 laborious than the expansions of integral powers required for 

 simple ascending division ; but, as a matter of fact, the labour 

 is avoided by calculating the invert from the algebraic series 

 which can easily be obtained by observing the law of formation 

 of the coefficients of the successive powers of in the quotient — 

 as already shown for simple ascending division in Section III (3). 

 This law may be readily worked out by the reader in the case 

 of the general trinomial equation x n + px"~ r = y, from which it 

 will be seen that the coefficients of the invert of o" + po n ~ r 

 reduce to the ordinary binomial coefficients multiplied by a 

 factor. Both this and the general cases have been discussed 



