VOL. XXIII.] PHILOSOPHICAL TRANSACTIONS. 663 



of the cissoid revolved about the diameter of its generating circle^ is exhibited 

 in finite terms, having given the quadrature of the hyperbola. 



Theorem VI. Let a be the area of a curve whose absciss is x, and ordinate 



_? — ; and let b be the area of another curve having the same absciss x, but 



rr + XX 



its ordinate : then will the area a = — + -— — r ^c. + ?-"'b. 



rr + xi m — l m—3 m — 5 



Corol. If TO be expounded by any term of the following series, O, 2, A, 6, 8, 



&c ; then the quadrature of the curve whose ordinate is :i:^rr^^ depends on the 



rectification of the circular arc. For if with the centre c, and radius ca = ?■, 

 the circle aeg be described, fig. 4, pi. 14 ; then drawing the tangent ak = x, 

 and joining ck meeting the periphery in e ; then the arc ae divided by rr, 



is equal to the area of the curve, whose ordinate is — , — . 



T ' rr + XV 



j1 general Corollary to these six Theorems. 



Every mechanical curve, whose quadrature depends on any of the infinite 

 number of curves, whose ordinates can take the following forms, viz. 



X"' X" 



x'^Vdx + XX ; ~7T^r^\ x'^Vrr + xx; 'J=z^=-, -rir; — r— ? may be squared 

 — Vdi±xx "~ v;;-i.r.t d±x rr + xx •' ' 



by these series. One example of which may suflSce. 



Supposing the cube of the circular arc, corresponding to the versed sine, be 

 made the ordinate of a curve, whose absciss is the same versed sine ; to find the 

 area of that curve. 



Let the absciss be ,r, the circular arc v ; then the fluxion of the area is v^x. 

 Suppose the area to be v^x — q. Then v^x + 2 v' vx — q =■ v^x, whence q = 

 3 v^vx. But V = 



— ■ , theref. 0= — y-. — ^ ; but by th. 2, , = , — y = v — 



HVdx—xx ^ 2vdx—xx ■' Vdx — xx 2Vdx—xx -^ 



y; hence q = ^di'v — f dv"y, therefore q =■ \ dv^ — fl. 4 dv^if. Hence it is 



reduced to this, that we must find the fluent quantity of the fluxion 4 dv-y. 



Let it be the quantity \ dvhj — r. 



Hence 4 dv^-y- + 3 dvvy — r = -| dv-y ; 



Therefore r = 3 dvvy = J d'vx. Let r = -| d'vx — s ; 



Then a d'^vx = 4 d'vx + -f d-xv — s ; 



Therefore 5 = 4- d'xv = . , " ^ = ^ d^v — 4- d^y by theorem 2. 



AiVdx—xx 



Hence j = -f- d^v — -f d^y; and therefore the area sought = 

 v^x — \ dv^ + i dv-y — 4 d^vx -^ %d?v — -f- d^y. 



Now because solids generated by the rotation of curves, with the superficies 



