Elements of analysis as applied to the mechanics of engineering and machinery . ?-, we term the determined portion P T of the line of con-tact between the point of contact P and the point of intersection T Fig. with the axis of abscissas, the tangent^ and its projection TM inthe axis of abscissas, the suhtaiigent^ and we thus havesubtang, = P 31 cotang. P TM = y cotang, a = y for example, for the parabola: suhtang. dy y — == 2 ^. y Here, therefore, the subtangent is equal to twice the abscissa, andthe position of the tangent for every point P of the parabola is,accordingly, easily
Elements of analysis as applied to the mechanics of engineering and machinery . ?-, we term the determined portion P T of the line of con-tact between the point of contact P and the point of intersection T Fig. with the axis of abscissas, the tangent^ and its projection TM inthe axis of abscissas, the suhtaiigent^ and we thus havesubtang, = P 31 cotang. P TM = y cotang, a = y for example, for the parabola: suhtang. dy y — == 2 ^. y Here, therefore, the subtangent is equal to twice the abscissa, andthe position of the tangent for every point P of the parabola is,accordingly, easily indicated. For a curved surface BCD Fig. t, the angles of inclination aand /5 of the tangents P T and P C/ at a point P are determined bythe formulae: tanq. a = —- and tana. (3 = ^-. ox cy The plane P TV laid through P T and P CTis a tangential planeof the curved surface. Art. *!. For a function y =i a -{- ^mf (^), we have: dy = \_a ^ mf {x -|- 8^?)] — [«- + m/ {x)\?=i a — a -\- mf {x -{- dx) — mf (x)= m [/ (x J^dx) —f {x)2; i. e.: L . . . d [a-\-mf (x)2 = 7ndf (x) ]: 8 (5 4- 3 ^^) = 3 [(x + dxy — ^2] = 3 . 2 xdx = is likewise: d(i — ^x)= — ^d (xf = — ^[(x-^ dxy —x2 = —
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