Page:The method of fluxions and infinite series.djvu/33

by multiplying into $y^{3}$ there would ariſe $xy^{3}=a^{2}y^{2}-2a^{3}y+3a^{4}$. And ſo of others.

27. Thirdly, when the Equation is thus prepared, the work begins by finding the firſt Term of the Quote; concerning which, as alſo for finding the following Terms, we have this general Rule, when the indefinite Species ($x$ or $z$) is ſuppoſed to be ſmall; to which Caſe the other two Caſes are reducible.

28. Of all the Terms, in which the Radical Species ($y$, $p$, $q$, or $r$, &c.) is not found, chuſe the loweſt in reſpect of the Dimenſions of the indefinite Species ($x$ or $z$, &c.) then chuſe another Term in which that Radical Species is found, ſuch as that the Progreſſion of the Dimenſions of each of the fore-mentioned Species, being continued from the Term firſt aſſumed to this Term, may deſcend as much as may be, or aſcend as little as may be. And if there are any other Terms, whoſe Dimenſions may fall in with this Progreſſion continued at pleaſure, they muſt be taken in likewiſe. Laſtly, from theſe Terms thus ſelected, and made equal to nothing, find the Value of the ſaid Radical Species, and write it in the Quote.

29. But that this Rule may be more clearly apprehended, I ſhall explain it farther by help of the following Diagram. Making a right Angle BAC, divide its ſides AB, AC, into equal parts, and raiſing Perpendiculars, diſtribute the Angular Space into equal Squares or Parallelograms, which you may conceive to be denominated from

$x^{4}$	$x^{4}y$	$x^{4}y^{2}$	$x^{4}y^{3}$	$x^{4}y^{4}$
$x^{3}$	$x^{3}y$	$x^{3}y^{2}$	$x^{3}y^{3}$	$x^{3}y^{4}$
$x^{2}$	$x^{2}y$	$x^{2}y^{2}$	$x^{2}y^{3}$	$x^{2}y^{4}$
$x$	$xy$	$xy^{2}$	$xy^{3}$	$xy^{4}$
$1$	$y$	$y^{2}$	$y^{3}$	$y^{4}$

the Dimenſions of the Species $x$ and $y$, as they are here inſcribed. Then, when any Equation is propoſed, mark ſuch of the Parallelograms as correſpond to all its Terms, and let a Ruler be apply'd to two, or perhaps more, of the Parallelograms ſo mark'd, of which let one be the loweſt in the left-hand Column at AB, the other touching the Ruler towards the right-hand; and let all the reft, not touching the Ruler, lie above it. Then ſelect thoſe Terms of the Equation which are repreſented by the Parallelograms that touch the Ruler, and from them find the Quantity to be put in the Quote.

30. Thus to extract the Root $y$ out of the Equation $y^{6}-5xy^{5}+{\tfrac {x^{3}}{a}}y^{4}-7a^{2}x^{2}y^{2}+6a^{3}x^{3}+b^{2}x^{4}=0$, I mark the Parallelograms belong-