SUPERIOR AND INFERIOR LIMITS TO THE ROOTS 
OF AN EQUATION. 509 
Thus, for instance, if there be two sequences so that 
and 
the g-reatest and least roots of x deduced from these equations will be superior and 
inferior limits respectively to the roots of/x; from which it is clear that if leaving 
all the other equations unaltered, except those which contain respectively ql and 
we write in place of these 
the roots of the system of i-{-i equations thus modified will a fortiori be limits to the 
roots oifx, but then the quantities 
e+~, •',+i, 
form the same single series as would correspond to the two sequences 
5'i §'2 • • • ^i+ 1 . • . 5'i+i'} 
treated as a single sequence, and the same is obviously the case for any number of 
sequences*. 
Art. {v.). If we consider a single sequence as q^.-.q^, and write 
q,=zafx—c,) q2 = a^{x—c^)..,q^=afx—c^) 
where a„ , a„ are supposed to have all the same sign, and write 
aj(x- al{x- c,)>= ..al(x- c.Y= (fj ‘ 
It follows from this, that if jj, ...g^ be all linear functions of j?, and if 
Q=(??-P?)(?2-(/^2+^) yj ) (gl- 
no root of Q can lie between the extreme roots of the function K, used to denote the cumulant 
+ 
the square roots being understood to be taken so as to make the sign of the coefEcients of o' all of them positive ; 
and from a preceding article we know that either extreme root of Q can be made to coincide with a corresponding 
extreme root of K. Hence we have an d priori solution of the following question, viz. “ To determine the (» — 1 ) 
positive quantities /t,. p„_ 1 , so as to make the greatest root of Q a minimum and its least root a maximum ; ” 
for the greatest root of K will be the minimum greatest root of Q, and the least root of K the maximum least 
root of Q. Calling these respectively I and X, the two systems of values of p„ p„...pn-i required will be 
obtained by substituting respectively I and X for x in the equations 
1 
1 
1 
