236 SCIENCE PROGRESS 



same combinations of c, d, e, etc., with the same signs, but that 

 these combinations are generally affected by different numbers 

 in the two cases (" Verb Functions," p. 53). 



N. In order to facilitate division of (/> by — h when h is 

 a decimal fraction, first shift the decimal point suitably both 

 in the dividend (f> and in the divisor — h, and then shift it 

 backwards in the quotient. For instance (Example D), in 

 order to divide o s — 70 + 7 by — 1*35, carry out the division 

 of o* — 700000 + 7000000 by — 135, giving the quotient 

 o* — 4050* — 153250 -f- 10375. Then shifting back the decimal 

 point, we have o' — 4*050' — 1*53250 + '010375 as the actual 

 result. 



Show also that to divide operatively by — 1*35 is equiva- 

 lent to dividing operatively by — 1, — 0*3, an d — 0*05 

 successively in any order we please. 



(6) In subsequent sections I hope to show by the geometrical 

 interpretation of the series that it is the algebraic expression 

 of the arithmetical processes of Dary and Newton (and of a 

 general theorem of my own), and to give tests for convergence ; 

 to prove that the series is nothing but the first term of the 

 multinomial theorem of operative volution or iteration ; and 

 to examine descending division, " critical points," and the 

 selection of the appropriate " setting " of the original equation 

 for each root in turn. The general series, which contains all 

 the roots together and is the true algebraic invert, will remain 

 to be considered. 



Alexandria, 



August 15, 1915. 



