164 Proceedings of the Royal Society 
which shows that the fractions 
11 69 509 3937 30901 243657 
2-3’ 2-ll’ 2-69’ 2-509’ 2-3937’ 2-30901’ 
converge to the greatest root of the equation. Or, performing the 
division from the right hand, observing to change the signs, we 
have 
•p &c. -p 
898034 
36 
a;6 + 
91913 
3'5 
+■ 
9367 
34 ^ 
964 . 103 
33 
32 
showing that the fractions 
3-91913 3-9367 3-964 3-103 3-13 
•’ 898034 ’ 91913 ’ 9367 ' 964 ’ 103 ’ 
converge to the least root of the equation. 
In this case we have the duserr progression, 
„ 9367 964 103 13 „ H 69 509 3937 „ 
34 ’ 33 ’ 32 ’ 3 ’ 2 ’ 22 ’ 23 ’ 24 ’ 
of which the centre term 3 is the index of the order of the equa- 
tion, and of which if we take any four consecutive terms, which 
we may denote hy P, Q, E, S, we have the equation 
2S«-11E + 13Q-3P = 0, 
by help of which we can readily continue the progression either 
way; thus, 
g 3P-13Q + 11B ^_ 3Q-13E + 11S 
for the progress to the right hand ; or. 
&c., 0 = 
13P-11Q + 2E 
3 
p__ 13Q-llB + 2S 
without the necessity of going through the details of the division. 
If we denote hy , r^, r^, &c., the roots of the equation, the co- 
efficients of the powers of x in the duserr progression are the sums 
of the powers of &c. inverse of the power of x ; that is to 
say, 
(r^) ; B = ^ (r^) ; C = 2 (r®), &c. ; while 
a = 2 (r~ 1) ; (3 = '^ ; y = (r “3)^ &c. 
Now if we have a number of unequal quantities, , r^, r.^, and take 
their successive powers, the power of the largest of them may be 
made to exceed the sum of the corresponding powers of all the 
