l6 PHILOSOPHICAL TRANSACTIONS. [aNNO '16Q5. 



hence x = z" — a, and i- = "2 2 ir ; therefore , = 2 z'^ i — 2 a z, and hence 



Vx + a 

 ^z^ — 1 uz or ^x — ^ a ^ X -{■ a will be the area sought. 



But it often happens that some curves, such as the circle or hyperbola, are 

 of such a nature, that it would be in vain to endeavour to free their fluxions 

 from surds, in which case the value of the ordinate must be reduced to an infi- 

 nite series ; then every term of this series being multiphed by the fluxion of the 

 absciss, as above, the fluent of every term must be separately found, and the 

 new series thus arising will exhibit the quadrature of the curve proposed. 



With the same ease may this method be accommodated to the measures of 

 solids formed by the rotation of a plane, viz. by assuming for their fluxion the 

 product of the circular base into the fluxion of the absciss. Let the ratio of 

 a square to its inscribed circle be 1 to n ; then the equation to the circle being 

 y y z= dx — XX, therefore 4 ft. d x x — x"^ x is the fluxion of a portion of the 

 sphere, and consequently 4 n.-f d x^ — -3- x^ is the portion itself. But the cylin- 

 der circumscribed about this is 4 n.dx- — .r^ ; therefore the ratio of the portion 

 of the sphere to the circumscribed cylinder, is as -i- c^ — -^ x io d — x. 



The rectification of curves will be obtained if the hyf>othenuse of the right- 

 angled triangle, whose sides are the fluxions of the absciss and ordinate, be 

 considered as the fluxion of the curve ; observing that, in the expression of 

 that hypnthenuse, only one of the fluxions be retained, and only the indeter- 

 minate quantity of the same ; as will be plain from the examples. 



From the right sine b c, (fig. 2, pi. 1) of the arc a c, being given, to find the 

 arc. Putting A b = x, B c = j/, o a = r ; let c E be the fluxion of the absciss, 

 D E the fluxion of the ordinate, and c d the fluxion of the arc ac. Now from 

 the property of the circle, 1 r x — x x = y y; hence 1 r x — 2 x i' = 2 y i/, and 

 X = -^-'^ ; but c D = w y 4- x x z=. 11 11 A 4-^ = y u A — i-^y— = 



r — X ^ ^ ' -^-'rr — Irx-'rxx ^ ~j ' j- y — y y 

 S1JJ_ therefore c D = . -^ ; but . -'' is the product of .- or 



r r — y y V r r — y y v ;■ r — y y Vr r —j y 



r r — y y\~^ into 7;/: hence i f r r — y y\~ he thrown into an infinite series, and 

 every term multiplied by r y, and the fluent of every product be found, there 

 will be obtained the length of the arc a c. 



In like manner may the arc be found, from the versed sine being given. 

 Thus, resuming the equation 2 /• x — 2 x x = ■! y y, it gives y = — ^ — ; but 



. . , . . . . , rrxx — 2rxxx x- x x . ■ , r r i x — '2 r x x x + x- x 

 C D= X X A- 1/ 1/ = X X A- = X X A ;r , 



or, reducing all to the same denominator, and expunging the p.irts which cancel 



each other, it is = „ , and hence c u = -^\ then, from wliat has 



1rx — xx ^/^Irx-xx 



been done above, the length of the arc will be easily obtained. 



