394 SCIENCE PROGRESS 



multinomial theorem. The following examples will make 

 this clear : 



A. a* 3 — 2x — 5 = o 



Newton's equation. In Section III (5) Example A we obtained 

 the single real root of this by simple ascending division, but 

 only after shifting the origin by dividing by — 2. Now write 

 the equation in the form 



that is, 

 and 



Carrying out the division to three terms, 



2 8 



o 3 — 20I0 |"o J + -o _i — -x-o -1 + . 



3 81 



2 _, 4 _, 40 - . 



o * — -0 — jr-o 6 + • . • 



3 9 81 



2 _i . 4 _ 3 , 40 _ 5 



-0 + ^0 + ~o ° - . . 

 3 9 81 



2 _j , 2 2 - . 2 8 _ 5 . 



-O -\ 3 + — + . 



3 33 39 



8 5 

 5 — . 



81 



Applying the operator in the quotient to the subject 5, we 

 have 



* = 54 + 3 5 ^~87 rS + '-- 

 = 1*70998 +0*39000 —0*00677 = 2*09321 . . . 



The root is 2*09455 • • • J so that we have obtained three figures 

 of it correctly from only three terms of the invert and without 

 any transformation. 



B. x* — $x + 1 = o 



This is tjie " centred " form of y % — 3V 2 — 2 v + 5 = o, dealt 

 with in Section III (5) Example B. It has three roots, of 

 which we will now obtain two at once by a single descending 



