Page:The method of fluxions and infinite series.djvu/29

12. Also the Order of the Terms may be inverted in this manner $xx+aa$, in which case the Root will be found to be $x+{\tfrac {aa}{2x}}-{\tfrac {a^{4}}{8x^{3}}}+{\tfrac {a^{6}}{10x^{5}}}-{\tfrac {5a^{8}}{128x^{7}}}$ &c.

13. Thus the Root of $aa-xx$ is $a-{\tfrac {xx}{2a}}-{\tfrac {x^{4}}{8a^{3}}}-{\tfrac {x^{6}}{10a^{5}}}$ &c.

14. The Root of $x-xx$ is $x^{1/2}-{\tfrac {1}{2}}x^{3/2}-{\tfrac {1}{8}}x^{5/2}-{\tfrac {1}{16}}x^{7/2},$ &c.

15. Of $aa+bx-xx$ is $a+{\tfrac {bx}{2a}}-{\tfrac {xx}{2a}}-{\tfrac {b^{2}x^{2}}{8a^{3}}},$ &c.

16. And ${\sqrt {\tfrac {1+axx}{1-bxx}}}$ is ${\tfrac {1+{\tfrac {1}{2}}ax^{2}-{\tfrac {1}{8}}a^{2}x^{4}+{\tfrac {1}{16}}a^{3}x^{6},{\text{ }}\&{\text{c.}}}{1+{\tfrac {1}{2}}bx^{2}-{\tfrac {1}{8}}b^{2}x^{4}+{\tfrac {1}{16}}b^{3}x^{6},{\text{ }}\&{\text{c.}}}}$ and moreover by actually dividing, it becomes

$1$	$+{\tfrac {1}{2}}bx^{2}$	$+{\tfrac {3}{8}}b^{2}x^{4}$	$+{\tfrac {5}{16}}b^{3}x^{6},$	&c.
	$+{\tfrac {1}{2}}a$	$+{\tfrac {1}{4}}ab$	$+{\tfrac {3}{16}}ab^{2}$
		$-{\tfrac {1}{8}}a^{2}$	$-{\tfrac {1}{16}}a^{2}b$
			$+{\tfrac {1}{16}}a^{3}$

17. But these Operations, by due preparation, may very often be abbreviated; as in the foregoing Example to find ${\sqrt {\tfrac {1+axx}{1-bxx}}}$, if the Form of the Numerator and Denominator had not been the same, I might have multiply'd each by ${\sqrt {1-bxx}}$, which would have produced ${\tfrac {\sqrt {\begin{aligned}1&+ax^{2}-abx^{4}\\&-b\end{aligned}}}{1-bxx}}$, and the rest of the work might have been performed by extracting the Root of the Numerator only, and then dividing by the Denominator.

18. From hence I imagine it will sufficiently appear, by what means any other Roots may be extracted, and how any compound Quantities, however entangled with Radicals or Denominators, (such as $\left.x^{3}+{\tfrac {\sqrt {x-{\sqrt {1-xx}}}}{\sqrt[{3}]{axx+x^{3}}}}-{\tfrac {\sqrt[{5}]{x^{3}+2x^{5}-x^{3/2}}}{{\sqrt[{3}]{x+xx}}-{\sqrt {2x-x^{2/3}}}}}\right)$ may be reduced to infinite Series consisting of simple Terms.

Of the Reduction of affected Equations.

19. As to affected Equations, we must be something more particular in explaining how their Roots are to be reduced to such Series as these; becauae their Doctrine in Numbers, as hitherto deliver'd by Mathematicians, is very perplexed, and incumber'd with superfluous Operations, so as not to afford proper Specimens for performing the Work in Species. I shall therefore first shew how the