OPERATIVE DIVISION 401 



the roots of an equation once, twice, thrice . . . together are 

 successively equal to the successive coefficients of the equation 

 with their signs alternately changed and not changed. The 

 demonstration requires the use of the symmetrical functions 

 of the roots and is laborious but elegant. It is given for the 

 general equation in " Verb Functions," pages 46 and 60-63. 

 Hence the n series obtained by substituting the n values of l/y 

 in the operative series for yfr' 1 are actually all the n roots of the 

 equation. The series for ^r~ l {when yfr n possesses no negative 

 powers of 0) is therefore the complete invert of yfr lt . It should be 

 added that I have not yet found how to recover more than one 

 or two numerical roots from this series : it generally furnishes 

 one ; but the remaining real roots must be got from the 

 secondary inverts obtained from the equation divided by 

 various powers of x, as shown in Examples B, C, D of ( 1 ) above ; 

 or from " vicarious functions " (Part III). 



K. Prove that [[>„][o + p]]~ l = f~ n l -p- 



On dividing yjr n operatively by o —p, we obtain yfr n (0 +p) f 

 which is yjr n when the origin has been shifted. Inverting this, 

 we find that p vanishes from the coefficients of all the powers 

 of in the invert except from that of o°. These coefficients are 

 therefore invariants ; but from this point the study of ty- 1 

 takes a direction which does not concern us at the moment. 



L. Splitting the Subject. 



This is often useful for arithmetical work. For example, the 

 complete invert of x % — 2x = 5 has for subject the cube root 

 of 5. This being troublesome, we may take the equation in the 

 form x 5 — 2x + 3 = 8 — the invert of which is a function of 

 ^/8, that is, 2. We now have to invert, not o 3 — 20, but 

 o 3 — 20 -f- 3, which, however, is nearly as easy by use of the tabu- 

 lated series. If the original subject was %/S, the convergence 



Vr 



is quickest if the new subject is V 5 H — f'X — X" , where X is 

 the root and n is the degree of the equation. In this example 

 the new subject ought to be about V — , if we take X = 2 ; but 



the subject V -Z will be nearly as quick and easier to work with 

 — quicker than \^8 and nearly as easy. 



