OF THE REAL ROOTS OF TWO ALGEBRAICAL EQUATIONS. 487 
same; and it is evident that the effect upon the sign of the Indicatrix at the end of 
eveiy such interval depends, not upon the number of pj’s grouped together in such 
interval, but upon the form of the group as regards its being made up of an odd or 
even numbei of terms [the first interval will of course be understood to extend from 
-I- CO to a value immediately inferior to h„ and the last from a value immediately 
inferior to hp to —co]. Hence as regards the relation of the signs of the Indicatrix 
at the beginning to the sign at the end of every such interval, nothing will be altered 
by taking away any even number of tj's that may be found therein. If we suppose 
this to be done, we shall then have in some of the intervals one occurring and in 
the other intervals no ti ; that is to say, some of the A’s will be separated by single ^’s, 
but other h s will come together. Again, by removing any even number of A’s not 
separated by rj^ (and thus removing an even number of intervals), it is clear timt as 
many changes of sign of the Indicatrix will have been done away with from + to — 
as from — to +, and no effect upon the excess of the one kind of changes of sign over 
the other kind of changes of sign will have been produced. By removing pairs of A’s 
in this manner, it may happen that ri's will again be brought together, any even number 
of which, not separated by A’s, may again be removed and then pairs of s not sepa- 
lated by s in their turn, and so continually toties quoties until at length we must arrive 
at a reduced system of A’s and ns, where no two h's and no two n^ come together, or 
else all the h's and all the n^ will have disappeared. Let the scale of Ks and n^ thus 
simplified and reduced be called the effective scale of intercalations. The number 
of h's and the number of ns in any such scale will be equal, or will at most differ 
from one another by a unit, since at each part of the scale, except at the end, every 
h is followed by an n and every n by an h. If the scale begins and ends with an h, 
there will of course be one more h than n ; if it begin and end with an n, there will be 
one more n than h ; if it begin with an h or an n and end with an n or h, there will 
be as many of the one as of the other. 
1st. Suppose the effective intercalation scale to commence with an A; then in passing 
from +CO to just beyond the first h the sign of the indicatrix^ changes from + to - ; 
It changes again from _ to + as it passes the first n, then again from + to - as it passes 
the second h, and so on ; that is to say, there will be a change always in the same direc- 
tion from -b to — as x passes, from being just greater than to being just less than any h 
appearing in the effective scale. 2nd. If the effective scale begin with the indicatrix 
will conversely be negative after passing the first and every subsequent n, and change 
from - to + in the act of passing through the first and every subsequent h. So 
that on either supposition the changes of sign for the effective scale always take place 
m the same direction, and the number of h's in the effective scale will be measured 
by the number of such changes, and consequently will be measured by the difference 
between the number of times that the indicatrix^ changes its sign from -f- to 
as a: passes through each in turn of the real roots of fx, and the number of times that in 
passing through any such root it changes its sign from - to -f ; if the former number be 
MDCCCLIII. q „ 
