165 
of Edinhurgh^ Session 1863-64. 
others in any required proportion, so that if K denote the greatest 
of all the roots, the ratio : % (r™) may he brought, by taking m 
sufficiently great, as nearly to a ratio of equality as we desire ; of 
necessity, the ratio E^^+i ; % (r”*+i) will be still more nearly that of 
equality ; wherefore the quotient 
% (7'^) 
may be made to approximate to the value of K within any pre- 
scribed degree of nearness. 
Similarly, on taking the successive inverse powers, those of the 
greater diminish much more rapidly than those of the smaller 
quantities ; wherefore we may continue the progression towards the 
left hand until the sum 2 (r~"^) may have to the - power of the 
least root, p, a ratio differing from that of equality as little as may 
be desired, in wffiich case the two ratios 
^ ^ and : p~"» 
will be nearly alike ; that is to say, the quotient 
(r-^) 
S (r — — 1) 
will 
differ from the smallest root, p, by an imperceptible quantity. 
The chief use of this a23proximation will probably be found in 
practice to be to give a starting point for the more rapid methods 
already known, which methods are only rapid when the root has to 
be approached from a small distance. Yet, having obtained a 
pretty close approximation, s, to some root, we have only to make the 
substitution x — s = y, to obtain a new equation, having one root 
very small in comparison with the others, and to which, therefore, 
the approximation by this method will be very rapid. 
When the root E is positive, the signs of the quantities A, B, C, 
&c. become continuous and -f ; if E be negative, they must be 
alternate ; and it is to be observed that if the coefficients of the 
even powers of x become negative, the root must be imaginary. 
Also, if there should be two roots, E and - E^, nearly equal to 
each other, the coefficients of the odd powers will become small in 
comparison with those of the even powers, in which case it may be 
convenient to take 
as the formula for approximation. 
VOL. V, 
Y 
