36 PHILOSOPHICAL TRANSACTIONS. [aNNO 1703. 



Example 2. Let z' -f aif = bzy^. Here m^.'j, 72 = 3, e=l, r = 2; 

 which give / = 2 ; therefore, by note 4, the first three terms of the series give 



the required area = j^ zy - TJ^ ^ - ^^^^ ^ V . 



Example 3. Let z^ + ky^ = kz' "^y^K Here wi = 3, w = 5, e = 2, r = 1 1 ; 

 but because these do not make / an affirmative integer number ; therefore, by 

 the 2d rule, free the term hz -^^" from the quantity z ; then the equation be- 

 comes z* — hy^^ = — hz^y^ ; where fl=— A, /?= — ^, m = 5, n=ll,e=2, 



5 k 



r = 5 J which give / = 1 ; hence the area = -^ zy — Tgi z^y ~ *• 



Example 4. Let z"^ ^ hy^ z= ^ ^^y* Here »i = 2, n = 2, e = 2, r = 2, 

 which do not make / an affirmative integer number; therefore freeing the 

 term — hz*y^ from the quantity z, then z° + ky"^ = Az - y ; where a = A, 

 A =r A, TO = O, n = 2, e = — 2, r = 2, which make / = 1 ; therefore the 



area = TZ~ y 



k 

 Example 5. Let z* — ^^ = — | z^ y*; where tw = 2, n = 6, e = 2, 



r = 4 ; which do not make / an affirmative integer number ; and the same thing 

 happens when either of the other terms is freed from z ; therefore, according 

 to the third rule, I seek the complement : thus, as before directed, making 



X z=z Y, and y = Zj the given equation becomes y^ -— ~ z* = — j z* y'' ; 



which, by rule 1, reduced to the general form, will be z'' — — y^ = — z* y''; 



where «» = 6, n=2, e = 4, r = 2; which do not make / an affirmative in- 

 teger number ; therefore, by rule 2, freeing the last from z, it becomes 



,s L Y* = — Z-* Y^; where to = 2, n = 2, e=— 4, r = 2; hence 



/ = 1 ; and a = , h = — ; then the complement of the required area is 



^jjY - ^ z-'' Y, or ^zy - ^ zy-^; and theref. flu. zy - -tzy + - zy-* 



IS the area sought. 



Sbct. n. Let 2* + ay" = bz^' y^ + " +/z*' y* "*■ " be an equation expressing 

 the relation between the ordinate z and absciss y. Then will the area be — 

 Azy + Bz« + y + '+ ez'« + y' + ^ + Dz^"' + y'^ + ^ + ez^ + '3/^ + ' + &c. 

 Where, putting 2c + n 3= r, and c + « = *, it will be 



A := ' ; B = • ==— A - » 



m + « i»i.c+ 1 +«.c+ I « 



w — ge-t.r.6A-4-w~<» .c + l + r.g-j-l./B-fgrf^ 



