468 ON A THEOREM CONCERNING EQUATIONS HAVING EQUAL ROOTS. 
In like manner eliminating ap, hq, cr between the second, third, fourth and filth 
equations, we have 
^ 1 1 1 
a h c _ 
^ I 
and so in general we have for all values of e, 
Ills, 
a h c _ 
¥ ¥ s,+2 
¥ ¥ ¥ s,+3 
whence it may immediately be deduced, that, upon the given supposition of there 
being only three groups of distinct roots, we must have the following infinite system 
of coexisting equations satisfied, viz. 
s^t+SiU+s^v+s^w^O say Lo =:0 
Li = 0 
62 f + '53M+'^4^+'^5'^ = ^ L 2 = 0 
+ = L 3 = 0 
.S 4 ^ + A' 5 ?« + 'S 6 «^ + "S7W^ = 0 , L4=0, 
&C. &C. &C. &C., 
and conversely, when this infinite system of equations is satisfied the roots must 
reduce themselves to three groups of equal roots. ^ 
Let now 9 be any function of ^2 ... which vanishes when this is the case 
Then 0 must necessarily contain as a factor some derivee of the infinite system of 
equations above written, i. e. some function of .^o ^ 2 , &c., which vanishes when 
these equations are satisfied ; L e. some conjunctive of the quantities Lo L, L, L 3 ; but 
it is obviously impossible in any such conjunctive to exclude from appearing, unless 
by introducing some other . with an index higher than a, and consequently ® cannot 
be merely a function of -^2 •^4 ^ 5 , nor consequently of n, and the first five coe u- 
cients ; or if such, it is identically zero, and so in general any function of n, and only 
2 i — 1 of the coefficients, which vanishes when the roots reduce to ^ groups of equa 
roots, must be identically zero, as was to be proved. 
Art. (ft.) It ought to be observed that the preceding reasoning depends essentia > 
upon the circumstance of w being left arbitrary. If » were given the proposition would 
no longer be true. In fact, on that supposition, the n roots reducing to /. distinct i oots 
would imply the existence of w-i conditions between the n roots ; and consequently 
n-i independent equations would subsist between the n coefficients, and tunctions 
could be formed of i only of the coefficients, which would satisfy the prescribed con- 
dition of vanishing when the roots resolved themselves into i groups of distinct 
identities. 
