Infinite Series . 107 
in order to inveftigate the area near the points of the abfcifs, 
where the ordinates become o or infinite, he transforms the 
equation, and finds feriefes exprefling the area near thofe points, 
in which feriefes the abfciffseor unknown quantities begin from 
thofe points. 
5. In the Meditationes it is afferted, that in a feries proceeding 
according to the dimenfions of x 9 if any root of the above- 
mentioned equations be fituated between the beginning of the 
abfcifs o and its end x 9 the feries will not converge : it is there- 
fore neceffary to transform the abfcifs fo that it may begin or 
end at each of the roots of the above-mentioned equations, and 
confequently where the ordinates become o or infinite, &c.; 
thofe cafes excepted where the ordinate becomes o, and its cor- 
refpondent abfciffa is a root of a rational function W of x 
without a denominator, and f WPa is equal to the given 
feries ; and in general the abfciflfe ought to begin from the 
above-mentioned points; for if they end, there the feries will 
converge very flow, if at all. 
6. It is further afferted, that if a and b 9 the values of the 
abfciffae between which the area is required, be both more near 
to one root of the above-mentioned equations than to any other, 
and n feriefes are to be found, whofe fum exprefs the area con- 
tained between a and b ; then that the fum of the ( n ) feriefes 
may converge the fwifteft, the diftances of the beginnings of 
each of the n abfciffae from the adjacent root will form a 
geometrical progreffion. 
7. Mr. Craig found the fluent of any fluxion of the for- 
mula (a + btf , + cx tn + 8 cc.y n x*- mJ x by a feries of the following 
P 2 kind 
