VOL. XXIir.J PHILOSOPHICAL TRANSACTIONS. ^5 



Section I. Let z"* + ay"" = ^z" y"" be an equation expressing the relation 

 between the ordinate z and absciss 3/; in which the exponents m, w, e, r, denote 

 any numbers, integer or fracted, affirmative or negative. Put r ^ nz=zc\ then 



wi 1 the area = — ; — zy -\ — — -— X ^ z' + 3/" + 



m-e.c+ l+r.e4- 1 bs^,, + .2^ + 11 fn-e.<2c+ 1 + r .^e + I fc_c^,^ + i j,Jc+ 1 

 '"wi.2c+l + ra.2e+lo*' •'' '»i.3c+l+«.3e+l a ^ 



-f &C. 



Concerning this series the following things are to be noted: 1. That the 

 capitals b, c, d, &c. denote the co-efficients of the terms immediately preceding; 

 2. That it exhibits the quadratures of all quadrible figures, whose curves are 

 defined by an equation of three terms; 3. And that these are always quadrible 



when —~ — ^^ is an affirmative integer number, which we may call b; 4. 



mn — mr — en ° ■' 



Particularly that / + 1 gives the number of the terms of the series, counted 

 from the beginning, that constitute the required area ; 6. That if we suppose 

 e = 0, this series will be changed into Newton's celebrated Binomial Theorem, 

 which theorem is therefore a particular case of this series ; 6. When application 

 is made of this series to any particular figure, these following rules are to be 

 observed; first, let the equation defining the given curve be reduced to the 

 general form ; then by comparing the particular equation with the general, let 

 the co-efficients a and h be found, as also the exponents tw, n, e, r. Secondly, 

 if the exponents thus determined do not make / an affirmative integer number, 

 according to the condition in note 3, then another term of the particular equa- 

 tion is to be freed from the quantity z ; and if the exponents again determined 

 do not give the condition of quadribility required, then the other term is to be 

 freed from the quantity z ; for all the three terms in the given equation cannot 

 by any means be freed from the quantity z. Thirdly, if the said condition of 

 quadribility does not belong to the equation, when treated according to the fore 

 going rule, then by the series find the complement of the area, or fluent of 

 yz ; which being found, the area required will become known : for it is well 

 known that zy — flu. ?/i = flu. zij. And that the complement may be obtained 

 by the series without confusion, in the given equation, defining the particular 

 curve, for z we may write y, and for y write z ; and having made this change of 

 the ordinate into the absciss, and the absciss into the ordinate, the equation 

 may be treated according to the precepts of the second rule, till the condition 

 of quadribility be obtained, or till it appear that no such condition can be had. 

 Example 1. Let z^ -f- y^ = bzy. Because here w = 3, n = 3, e=:J, 

 r = I, a = 1; therefore / = 1, and /+ 1 = 2. Then, according to note 4, 

 the first two terms of the series give the area =z .^zy — -^ bz^y ~ \ 



VOL. V. E 



