The Principles of Fluxions Designed for the Use of Students in the University
The Principles of Fluxions Designed for the Use of Students in the University
William Dealtry
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- 1 X = — . + hyp. Log. J 2x- X*, ^ 2X-X* J2X-X^ ^V ^ SI FLUENTS. 275 (344. ) Fluent 120. To find the fluent of z'lfy, where % is the arc of a circle, and y the sine. This fluent = ^^ - f. -^^ x z ; but ^ = /'^ _ I = y X 1 - "^ ' = (by the binomial theorem) ^ x : 1 + •2 — :Z + &c. And the whole fluent = -^ — — x : ~ 8r* w + 1 w+ 1 w + 2 + =L + r=^r + &C. w + 4. 2r' 7z+6. 8r^ (345. ) Fluent 121. To find the fluent of z^y, where z = the arc of a circle, and y the corresponding sine. Let T = the r...adius, and x the cosine ; then, by Art. 44, ry = xzj and yz= —rx. Assume v = z^i then, since the fluent of vy = vy-f. Yv, we have the fluent of z"y = zy -f. Y x nz'^'^z ; but yz=^rx; :, the second part =-^f. Nrz"-'x; and the fluent of this part = nz^'^rx -/. N. N-1, rxz'^'^z ; and since xz = ry, the fluent of the last term = - w . N- 1 . Z'^-^r'y + &c. Hence the required fluent = z^'y-^-n^'Wx-n . N— 1 . Z'^-^r^y - &c. (346. ) The same fluent may be found in the following manner : Assume t/x : «z" + fez"~' + C2;'*~^+ &c.
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