38 FHILOSOPHICAL TRANSACTIONS. [aNNO 1703. 



which shows the relation of the exponents m, n, e, c, and is this, viz. that 



KC — 2c + 2e — \c 4- T« -|- n . „ . , , . , ,, 



_ ^^_ ^^ be "'- affirmative integer number, which call /; but the 



other conditions respect the co-efficients, a, b, /, g, A, &c ; and lastly, / 4- 1 

 gives the number of the terms of the series, taken from the beginning, which 

 constitute the required area. 



CoroL From this general series may be deduced a series, exhibiting the qua- 

 dratures of figures, whose curves are defined by an equation consisting of any 

 terms, which constitute the general equation of the third section. For, to 

 obtain this, there needs only to be computed a series for an equation consisting 

 of as many terms of the general equation, counted from the beginning, as are 

 the terms contained in the equation defining the curves. Then from the values 

 of the quantities a, b, c, d, &c. the co-efficients byf, g, &c. may be expelled, 

 which do not belong to the equation proposed ; the others will give the area, 

 required, as will appear by an example. 



Sect. IV. Let z*" = ay"* + ^z*y*= + " + g^^'j/^" "*" " bE an equation expressing 

 the relation between z and y. Now because z"* = oz/" + ^z*y<= + " -hf^^^y"^" ^ " 

 + g^^' y^*^ ■*■ " is that part of an equation which, taking the terms in order from 

 the beginning, include the given equation; which hereafter, for brevity's sake, 

 may be called the complete equation ; therefore the areas of the figures, 

 whose curves are defined by the complete equation, will be = 

 Azy + Bz* + ' 3/*= + ^ + cz*« ■^^y^'' ^ ^ -\- Dz'* + ^y^*" + ^ &c : and the co-efficients 

 enter into the values of the quantities b, c, d, &c. If therefore in these values 

 there be put every where /= O, because the term /z**'^/**' +" does not enter 

 the given equation, we shall have the values of the quantities a, b, c, d, &c ; 

 which being substituted m the series, it will give the area sought. And cal- 

 culating from the beginning, it is found that 



m ^ e — m — c — n.A+m — eb^ c + n.e-\- 1 +m —e.c+ 1 Ab^ 



fn + n* m. c + 1 + « . e + l a* m . 2c -J- 1 + n . 2e + 1 « ' 



^ »i — 3e.l-A + 3c-|-nx — gA+OT — e.2c+l+ c-H».2e-f- 1 x *c. 



D — == == — f 



wo . 3c + 1 + n« . 3c + 1 



3g.c4- l+Sc-f n.t+ 1 X — gB + m — e.3c+ 1 -i-c-f-«.3e-f 1 x — *d . 

 ma . 4c + 1 -H no . 4c -H 1 



CT — 3e.2c+l-|-3c-|-n.2c + lX— gc-l-OT — <.4c-|-l-|-c-f«.4c-J-lx-ftE. 



F =:= ■ . __^_^______- , 



ma, 5c + I + na . 5e + 1 

 &C. 



From hence appears the progression in the rest in infinitum. And thus is ob- 

 tained a series exhibiting the quadratures of all the figures, whose curves are 

 defined by this equation of 4 terms, viz. z"» = ay" -|- ^z" y*= + *• -f gz^ y'* + *. And 

 it may be noted that the conditions of quadribility, and the number of terms 



