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

Rh by multiplying into $$y^3$$ there would arie $$xy^3 = a^2y^2 - 2a^3y + 3a^4$$. And o of others.

27. Thirdly, when the Equation is thus prepared, the work begins by finding the firt Term of the Quote; concerning which, as alo for finding the following Terms, we have this general Rule, when the indefinite Species ($$x$$ or $$z$$) is uppoed to be mall; to which Cae the other two Caes are reducible.

28. Of all the Terms, in which the Radical Species ($$y$$, $$p$$, $$q$$, or $$r$$, &c.) is not found, chue the lowet in repect of the Dimenions of the indefinite Species ($$x$$ or $$z$$, &c.) then chue another Term in which that Radical Species is found, uch as that the Progreion of the Dimenions of each of the fore-mentioned Species, being continued from the Term firt aumed to this Term, may decend as much as may be, or acend as little as may be. And if there are any other Terms, whoe Dimenions may fall in with this Progreion continued at pleaure, they mut be taken in likewie. Latly, from thee Terms thus elected, and made equal to nothing, find the Value of the aid Radical Species, and write it in the Quote.

29. But that this Rule may be more clearly apprehended, I hall explain it farther by help of the following Diagram. Making a right Angle BAC, divide its ides AB, AC, into equal parts, and raiing Perpendiculars, ditribute the Angular Space into equal Squares or Parallelograms, which you may conceive to be denominated from the Dimenions of the Species $$x$$ and $$y$$, as they are here incribed. Then, when any Equation is propoed, mark uch of the Parallelograms as correpond to all its Terms, and let a Ruler be apply'd to two, or perhaps more, of the Parallelograms o mark'd, of which let one be the lowet 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 elect thoe Terms of the Equation which are repreented 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 $$y^6 - 5xy^5 + \tfrac{x^3}{a} y^4 - 7a^2x^2y^2 + 6a^3x^3 + b^2x^4 = 0$$, I mark the Parallelograms belong- Rh