Infinite Seriei. 149 
ther applied in the fame Book to complicate irrational alge- 
braical functions of x, &c. 
Hence moft commonly the feries for the area contained be- 
tween two ordinates, or integral between two different incre- 
ments deduced by the common method will diverge ; on which 
account, in the fame Book, is given a method by interpolation 
of finding the area or integral contained between any two dif- 
ferent values of x by converging feries, if the area, &c. is finite. 
6. To find whether a given value ( + a) is lefs than the 
leaft affirmative or negative root (x) of a given algebraical 
equation A + Bx + Cx* + Dx 3 + &c. = o, if all its roots are 
portable ; transform the equation into another, whofe root s is 
the reciprocal of the root x- - of the given equation, and for 
% in the refulting equation write refpe&ively v + a and v — a% 
and if from the former fubftitution all the terms become ne- 
gative or affirmative, and from the latter they become alter- 
nately negative and affirmative, then will a be lefs than 
the leaft root of the given equation. If in the fame manner, in 
the given equation for x be fubftituted vfi-a and v — a, and 
the terms refult as before, then will (a) be greater than the 
greateft root affirmative or negative of the given equation. 
7. When the integral of an algebraical quantity, whofe in- 
crements are finite, is required ; firft, by the method given in 
Medit. Analyt. inveftigate the integral in finite terms, if it can be 
expreffed by them ; but if not, reduce it into infinite feriefes of 
which the integral of each of the terms can be found, and alio 
the feriefes for finding the integral contained between the two 
different given values of the variable quantity may converge. 
Seriefes of this kind have been given in the Medit. Analyt.- 
and innumerable of a like kind may be added for finding 
integrals 
