OPERATIVE DIVISION 403 



where «i, n 2 , n z . . . denote the binomial coefficients. Secondly, 

 retaining b and assuming B = b n , we evolve another series 

 the sub-coefficients of which contain powers of b in geometric 

 progressions (Part III). 



(4) The expansion of $1 is homologous with that of (<f> ) 

 and may therefore be called the operative multinomial theorem 

 or the multinomial iteration theorem. I do not know whether it 

 has ever been fully set out and discussed ; but it is fundamental, 

 covers most of the present subject, and contains many approxi- 

 mations, continued fractions, periodic functions, etc. It may 

 be obtained by other routes and put in other forms — but not 

 easily when <j> contains an absolute term. I will soon proceed 

 to show that the inverts given by operative division are nothing 

 but the first term of the expansion when <f> does contain an 

 absolute term — that is, are the first term when n = 00 of the 

 expansion of (p n (Section V (3, 4, 5)). 



These examples emphasise the extraordinary manner in 

 which the algebraic symbols meet all their obligations. The 

 permanence of algebraic form already notorious in the case of 

 ((f>) n is now observed to hold equally well in the case of [</>]". 

 So far as I can see there is no a priori reason for this permanence : 

 at all events it comes from the wonderful properties of the 

 binomial coefficients. 



By the use of we obtain a merely formal algebra — almost 

 an algebra of mere empty form. Questions of convergence 

 do not arise because expressions in o have no numerical value : 

 they commence only when we apply the operations to concrete 

 numbers — that is, fill in the empty moulds given by the former ; 

 as will next be indicated. 



V. (1) I now propose to approach the operative inverts 

 from quite another direction, and to give instances in which 

 they are evidently the algebraic expression of the arithmetic 

 methods of Dary and Newton for approximating to the roots 

 of numerical equations, or of the general procedure of which 

 these methods are special cases. I have space only to touch 

 upon the theme in this Part. 



The method of Michael Dary, 1 Philomath, was first an- 



1 See Mr. Walter Stott's article in Science Progress, October 191 5. I 

 regret that as this paper is being written on active service I cannot now give the 

 references to the literature. Dary's original example was more general than the 

 one which I use here— see Mr. Stott's article. — Alexandria, October i, 191 5. 



