VOL. XIX.3 PHILOSOPHICAL TRANSACTIONS. 203 



OB X ill- 3^ ^^^j g^ ^^^ 2. That if the exponent n beany positive integer number, 



or = O, or if 2n be an odd number, then the quadrature of the space abd is 

 expressed by a finite quantity, because of the series breaking off in these cases. 

 3. That q denotes the last term so breaking off. 4. That all those figures, in 

 which the series breaks off, have a portion geometrically quadrable, easily 

 assignable from the series itself: for if there be taken the absciss y = 



1 i_ 



r - " + ' X nq + ^1" + ', the arc belonging to this absciss will be geometri- 

 cally quadrable. 5. That only the irrational term ^ lay — yy is to be multi- 

 plied into the terms that follow it. 



Example 1. — Let z = v. Now because in this case it is r = 1, ?« = O, 



therefore =^=rrw" is the last term breaking off. Hence q = a, and abd = vy 



w + 11 -^ ° 



-~ av -\- a V2aij — y"-. Therefore if, by note 4, there be taken the absciss y = 

 a, that is, if the ordinate pass through the centre of the circle, the portion be- 

 longing to it will be geometrically quadrable : for then the area = aa, the 

 square of the radius. 



Example 2. — Let z = — . Because in this case r = -, n = \, there- 

 fore "^"^ — ^° v" ~ ' is the last term breaking; off, so that o = 4- a ; hence abd = 

 *r _2fv ^ y_+ja ^ 2ay _ yy ; and therefore if, by note 4, there be taken 

 y = »/ laa, the area belonging to the absciss will be geometrically quadrable, 

 and equal to '^ '/~Qa^ — i «' X v^ ^V a' + -i- a. 



Example 3. — Let z = ^. In this case r = — , n = 2, and therefore 



^ aa aa 



oA X 2n - ^ „ _ 2 jg ^i^g jggj. jgj.j^ breaking off; therefore ^ = f a. Hence the 



. r. .. . .,, , Grv' — \5aiv + ^ay'^ + ba^y + 15a3 x V2<2j/ — vv rpi „_^ 



mfinite series will be abd = -^ ^ ~_ — —. 1 here- 



18aa 



fore if, by note 4, there be taken y =■ 1/ \a\ the area belonging to this ab- 

 sciss will be geometrically quadrable, and equal to r^ X 



^'iay -yy. 



Secondly, let acf, fig. 2, pi. 5, be a parabola, its axis ae, vertex a, and 

 parameter = 8a. And let adg be a curve geometrically irrational, whose ordi- 

 nate BD cuts the parabola in c. Call the absciss ab = y, the ordinate bd = z, 



D D 2 



