5 88 SCIENCE PROGRESS 



/= o— {o 3 — bo 2 — co — (m+ g 3 ) — eo 1 — . . . }/{30 2 — 260 



c+ eo' 2 + . 



/ 



= _I + -^+(- C +- ^V 1 + j- (m 4- *») + ± bc + 

 3 9 \9 2 7 ' 13 27 



+ 4 £ 3 j Q -2 _|_/4 + A C 2 + A b 2 c _|_ 2 „ + 3 ) + 



^8i J [9 27 27 ' 9 v TS/ ^ 



+ -A. ^lo -3 +| 1 /+ - be +( lc +~ b 2 \m + g*) + 

 ^243 J [9 27 V9 27 / 



81 243 729 I 



/£=£ + - &+(-c + - b*)g- 1 +(-m+ - &c + A ^y 2 + 

 3 \3 9 / \3 3 27 r 



+ (- c+ 2 w+ -c*+ - £ 2 c + A &*V 8 + • • • ; 



\3 9 9 27 81 / a 



+ (l e -2-bd+±<*-±Vc-%- b*)g-* + . . . ; 

 \3 27 27 81 81 P 



which is right to four major terms. Some detail has been 

 given in order to show the important fact that Newton's iterand 

 generates precisely the same descending invert. (Care must 

 be taken with the term g z in /.) 



But we shall obtain the same result if we use only the first 

 term or terms of y}r' q in the denominator of the fraction in /. 



For mid-axial iteration we can take 



1 — + g — \/o 3 — bo 2 — co — m — co -1 — . . . 



=g + - 6+(-c+ - lAo^+f- m+ -be +£- b*)o- + 

 3 \3 9 / \3 9 81 / 



,( l -e+-bd + lc*+±p c + — lA>-*+ . ..; 

 +\3 9 9 27 243 ) 



/ 2 £=/i + 4) + /_ 1 +*_ 2 +/_ 3 4- • • •; 



which is right to five major terms. Or we may take Dary's 

 form : 



I = Jg* -j- # 2 + co + m + co 1 + 



= g+\g- 2 (bo 2 +co . . .) - V 5 (6o 2 +co . ..) 2 + 

 + g7£- 8 (fr> 2 +co...) 3 + ...; 



