234 SCIENCE PROGRESS 



(before operating on — and multiplying by iooo) invert it. In 



this case we invert o s + 4050 2 — 153250 + 10375. Dividing 

 numerically by 15325, we have, by the series, 



Remainder of root = '67700 + '01211 + '00046 + '00002 

 + etc., = '68959. 



Thus ultimately : 



x = 1-3568959 . . . 



But the two last figures are too low, as the terms of the 

 invert are all positive. Observe that as the tangential of this 

 function (after division by 15325) is about — 0*982 at its root, 

 0*6896, the convergence of its invert is quick ; while, as the 



tangential of the original function 1 — -\ — o 5 was — 0*210 



at its root 1*356896, the convergence of its invert was slow — see 

 (2) above. 



We gather generally from these examples that simple 

 ascending operative division may give convenient and quick 

 approximations in many cases ; but that in other cases, especi- 

 ally where there are equal or nearly equal roots, as in the last 

 example, the approximation may be very slow. In the latter 

 cases, however, it may be quickened by transference of the 

 origin to a point nearer the root. As already mentioned, it is 

 by no means pretended that this form of division is always the 

 best. 



E. Show that the coefficients of in the operative quotient 



</> 

 of •■ are also given by the successive remainders when <f> is 

 o — h ° J 



divided algebraically by — h once, twice, thrice, etc. 



F. Prove that if 



[0 + co l +do % + .. -]- 1 = + Co* + Do 1 + . . . 



by simple ascending operative division, then also 



[0 + Co 2 + Do' + . . .]-' = + co 2 + do 1 + • • • ; 

 and [0 + co l + do 3 + . . .][o + Co 2 + £>o 5 + ...]= 0. 



G. Prove by simple ascending operative division that 



r-?-i""=-e- 



Li + oj 1 — o 



Divide operatively by the algebraic quotient of 



o 



1 + 



