(= VP+P) = Y Y 3 = y X 1 ± - ; the fluent of which, universally expressed in an infinite series, n =y2 to. is y — — — 2n- lx2a2n.-.2 4n — 3 x 8 a4n--t &c. =6n —5 x 16 Case II. Let all the ordinates of the proposed curve A It M (fig. 16), be re ferred to a centre C : then putting the tangent IL P (intercepted by the perpen dicular C P) = t, the arch, B N, of a cir cle, described about the centre C, = x and the radius C N (or C B) = a; we have i : fi y (C R) : t (It P) ; and, con sequently, I = from whence the value of z may be found, if the relation of y and t is given. But, in other cases, it will be better to work from the following y equation, vie. z , • , Wawa is thus derived ; let the right line C R be conceived to revolve about the centre then since the celerity of the generating point R, in a direction per pendicular to C R, is to the celerity of the point N, as C B (y) to C N (a), it will therefore be truly represented by which being to (y) the celerity in the di rection of C lt, produced as C II (8) P (0, it follows that L_,: : : 8' : t'; a' whence, by composition, ? y2i,3 : : S + t' (e) ; therefore t" and consequently i V LL—u a' But the same conclusion may be more easily deduced from the increments of the flowing quantities : for, if R m, r m, and N n be assumed to represent (i, y, a'..) any
very small corresponding increments of A R, C R, and B N; then will C N (a) : C It (y) : : I: (the arch N n) : the similar arch It r And if the triangle It r a en (which, while the point m is returning back to R, approaches continually nearer and nearer to a similitude with C R B) be considered as rectilinear, we shall also obtain 11 R r m") = • NI/ •`.••• + e j and /- ' = (4" ' • as before. See Simpson's " Fluxions."