4 o3 SCIENCE PROGRESS 



For ascending division by Section III (2) we divide by — o* 

 and apply the quotient to a. Thus if X is the lesser positive 

 root 



X = a -f- a 2 -f 2a 3 + 5a 4 + i4« 5 + 42« 6 + 1 32a 7 . . . + remainder. 



But by iteration we have 



X = [0 + (a — + o 2 )]°# = [a + o~Yx . 



Hence x\ = a + # 2 



# 3 = a + (a + x y — a + ft 2 + 2«at 2 + #0* 

 # 3 = a +a 2 -\- 2# 3 +a 4 +2(« + a 2 ){2a -\-x 2 )x 2 +{2a +Ar 2 ) 2 #o 4 

 # 4 = a + a 2 + 2a 3 + 5a 4 + 4a 5 + 4« 6 + 4a 1 + a 8 + Z?*o 

 # 5 = a + a 2 + 2a 3 + 5a 4 + 14a 5 + 22a 6 . . . + a 11 + ifo 



where i?# means a definite number of terms in x . We see 

 at once that the first r terms in the value of x r agree exactly 

 with the first r terms of the operative series. If x = o, as it 

 may do in this case, Rx evidently vanishes, leaving a set of 

 terms beginning, in x 5 , with 22a 6 and ending with a lt , which 

 must be equal to the remainder of the operative quotient. But 

 it is not immediately clear why Rx should vanish for all appro- 

 priate values of x . 



Similarly for the equation o = a — x + ex 11 , we find that 

 the first terms of [a + c&'Yxo agree exactly with those obtained 

 in Section III (2) as the value of [0— co a ] -1 #. 



For the general case <£ = a — bo + co 2 — do 3 + . . ., we 

 shall find it laborious to obtain even the first few iterants ; but 

 have 



<£ 2 = a — b(a — bo + co 2 . . .) + c {a 2 — 2abo + (b 2 + 2ac)o 2 — 



- 2(bc + ad)o 3 . . .} - d{a 3 - za'bo + 3(ab 2 + a 2 c)o % . . .} + 

 + e{a i — 4a*bo . . .} — etc. 



</>o 3 = fl(i - b + b 2 ) + a s c(i - 3b + b 2 ) + « 3 {2c 2 (i -b) - 



- d(i -4b +3b 2 -b 3 )} +a*{e(i - 4b)-cd(s -6b) + 

 + c 3 } + . . .+ Ro. 



Now if <po = + /, then / = a - (1 + b)o + co 2 - do 3 + . . • ; 

 and to obtain the least positive root of / by ascending division 

 we employ the formula of Section III (3) and find, if B = 1 + b, 



r c • 1 d * i-i a a a. c iL 2 -i- /Hf! ^\^ 3 4- 

 L0 ~B°" + 5 "" J ' £"""£ + BB 2± \B 2 BIB^ 



. / c 3 cd . e \a 4 , , 

 + (55,-5^+5)2, +etc. 



