ON A THEOREM CONCERNING EQUATIONS HAVING EQUAL ROOTS. 467 
ances of k„ lt„ Ic,, k , ; and tlms with a slight degtee of attention to the preceding pro- 
cess the reader may easily satisfy himself that the preceding demonstration (although 
not so expressed) is in essence universal, and the form of r as an explicit function of 
X and of the roots o!f(x) is thus completely established for all values of m and of i. 
Supplement to Section III. 
On the Quotients resulting from the process of continuous division ordinarily applied 
to two Algebraical Functions in order to determine their greatest Common Measure. 
[Received October 20, 1853.] 
Art. {a.)* We have now succeeded in exhibiting the forms of the numerators and 
denominators of"^ developed into a continued fraction in terms of the differences of 
the roots and factors of/r. It remains to exhibit the quotients themselves of this 
continued fraction under a similar form. 
Lemma.— equation being supposed of an arbitrary degree n, there exists no 
function ofxi and of less than 2i of the coefficients f, which vanishes for all values of n 
whenever the n roots reduce in any manner to i distinct groups of equal roots ; or in 
other words, any function o/n and the first 2i-l coefficients of an equation of the nth 
degree, which vanishes for all values of n in every case where the roots retain only i 
distinct names, must be identically zero. 
To render the statement of the proof more simple, let i be taken equal to 3. And 
let the roots be supposed to reduce top roots a, q roots b, and r roots c. And let 5 , 
m general denote the sum of the rth powers of the roots. Then we have evidently ' 
p +q +r =So 
pa -\-qb +rc =s, 
pa^-\-qF-\-rc^=s^ 
pa^A-qF+rc^ — s^ 
pa^-Tq¥+rc*=s^, 
&c. &c., ad irfinitum. 
Eliminating p, q, r between the first, second, third and fourth equations, we obtain 
1 1 1 >^0 
a b c s^ 
F c^ 
a^ F c® a’3 
* The articles in this and subsequent sections to which Latin or Greek letters are prefixed, althou-h 
in strict connexion with the context, are supplementary in the sense of having been supplied since the date 
when Uie paper was presented for reading to the Royal Society. All the articles marked with numbers (from 
to 2), and the Introduction, appeared in the memoir as originally presented to the Society, June 16 18^3 
t In the proposition thus enunciated the coefficient of the highest power of ^ is supposed to be a numerical 
