4 o6 SCIENCE PROGRESS 



And if / does not conform to these conditions, a vicarious 

 operation F can always be found which has the same root or 

 roots as / and which does conform to them, either for a, ft, and 

 Xo or for other convenient limits and base, so that the iteration 

 of o + F will infallibly lead to X. 



Secondly, if o = fx has any number of real roots and is 

 continuous and begins positively, then the iteration of o +/ 

 will approximate to the first, third, fifth . . . roots in succession 

 if the conditions given above are complied with for each root 

 in turn ; and if not, vicarious operations F lf F 2 , F 3 . . . can 

 always be found which do comply with the conditions. And 

 similarly the iteration of o — / or of o — F will approximate 

 to the second, fourth, sixth . . . roots in succession — equal roots 

 being always separately counted, and the first root being the 

 least. 



While success is ensured if these conditions are complied 

 with, it is not always excluded if they are infringed, unless 

 they are infringed at the root itself. 



It is easy to see that the methods of Dary and Newton 

 are special cases of this theorem under the heading of " vicarious 

 operations." The operation o + 5 + 20 — o J is not itself 

 convergent because /', that is, 2 — 30 2 gives values lying be- 

 tween — 10 and — 25 when x lies between 2 and 3. But in 

 the form used above to illustrate Dary's method, we really took 



F = ^5 + 20 — 0, 



and x n = [o + ^5 + 20 — o]"a:o ; 



2 — * 



and as F' , that is, -(5 -f- 20) 3 — 1 gives values between — o'772 



and — 1 for all values of x, the iteration always succeeds, 

 whatever x may be. And F has the same roots as /. 



In Newton's method, F == —4f, which has the same roots 



/ /" 

 as / (even when these roots are multiple). Here F' =j J . J ~ — 1 . 



For Newton's equation, this gives values which increase from 



— ri2 when x = 2, to — when x = co; so that + F is 



3 

 always convergent when xq lies between 2 and 00 . The reader 



can explain why it is also convergent when x = 1 . Note that 



in this case, as always except when the root is multiple, F'x= 1 



