Page:The method of fluxions and infinite series.djvu/39

$y={\tfrac {1}{2}}a+{\tfrac {1}{2}}x+{\tfrac {x^{2}}{2a}}-{\sqrt {{\tfrac {1}{4}}a^{2}+{\tfrac {1}{2}}ax+{\tfrac {3}{4}}x^{2}+{\tfrac {x^{3}}{2a}}}}$; or more expeditiously by the Method of affected Equations deliver'd before, by which we shall have $y={\tfrac {x^{4}}{4a^{3}}}-{\tfrac {x^{3}}{4a^{4}}}*$, where the last Term required vanishes, or becomes equal to nothing.

43. Now after that Roots are extracted to a convenient Period, they may sometimes be continued at pleasure, only by observing the Analogy of the Series. So you may for ever continue this $z+{\tfrac {1}{2}}z^{2}+{\tfrac {1}{6}}z^{3}+{\tfrac {1}{24}}z^{4}+{\tfrac {1}{120}}z^{5}$, &c. (which is the Root of the infinite Equation $z=y+{\tfrac {1}{2}}y^{2}+{\tfrac {1}{3}}y^{3}+{\tfrac {1}{4}}y^{4}$, &c.) by dividing the last Term by these Numbers in order 2, 3, 4, 5, 6, &c. And this, $z-{\tfrac {1}{6}}z^{5}+{\tfrac {1}{120}}z^{5}-{\tfrac {1}{3040}}z^{7}+{\tfrac {1}{362880}}z^{9}$, &c. may be continued by dividing by these Numbers 2×3, 4×5, 6×7, 8×9, &c. Again, the Series $a+{\tfrac {x^{2}}{2a}}-{\tfrac {x^{4}}{8a^{3}}}+{\tfrac {x^{6}}{16a^{5}}}-{\tfrac {5x^{8}}{128a^{7}}}$, &c. may be continued at pleasure, by multiplying the Terms respectively by these Fractions, ${\tfrac {1}{2}}$, $-{\tfrac {1}{4}}$, $-{\tfrac {3}{6}}$, $-{\tfrac {5}{8}}$, $-{\tfrac {7}{10}}$, &c. And so of others.

44. But in discovering the first Term of the Quote, and sometimes of the second or third, there may still remain a difficulty to be overcome. For its Value, fought for as before, may happen to be surd, or the inextricable Root of an high affected Equation. Which when it happens, provided it be not also impossible, you may represent it by some Letter, and then proceed as if it were known. As in the Example $y^{3}+axy+a^{2}y-x^{3}-2a^{3}=0$: If the Root of this Equation $y^{3}+a^{2}y-2a^{3}=0$, had been surd, or unknown, I should have put any Letter $b$ for it, and then have perform'd the Resolution as follows, suppose the Quote found only to the third Dimension.