. A new treatise on the elements of the differential and integral calculus . 2,|,^ x = -{li —29), y- = 2j)x: whence ?/^ = ^^i^ y-=2^ {22jxy =z 2 )3 and I>^ = 27^ ^^ ~^) ^ ^ 27^ (^ -^) for the required equation. If the origin of co-ordinates be transferred to a point atthe distance p in the direction of positive abscissae, the newbeing parallel to the primitive axes, the equation of the evo-lute takes the form 27j9 ^ or V rt We readily recognize that this curve is symmetrical with re-spect to the axis of abscissse, and that it extends without limitin the direction of x differenti
. A new treatise on the elements of the differential and integral calculus . 2,|,^ x = -{li —29), y- = 2j)x: whence ?/^ = ^^i^ y-=2^ {22jxy =z 2 )3 and I>^ = 27^ ^^ ~^) ^ ^ 27^ (^ -^) for the required equation. If the origin of co-ordinates be transferred to a point atthe distance p in the direction of positive abscissae, the newbeing parallel to the primitive axes, the equation of the evo-lute takes the form 27j9 ^ or V rt We readily recognize that this curve is symmetrical with re-spect to the axis of abscissse, and that it extends without limitin the direction of x differentiation, we find dv _S rj^ cV-r _3 \~T~ _i_ 1 1 Therefore, at the origin of co-ordinates, the axis of x is tangentto the curve, and this point is a cusp; and, since the sign of y-2 is the same as that of v^ the curve is at all points convextowards the axis of ic. 183, The expression forthe radius of curvature andthe equation of the evoluteof the hyperbola may be de-^ duced from those for the el-ii lipse by changing h^- into— 5^. Thus we have, for theradius of curvature,. CURVATURE AND EVOLUTE OF CYCLOID. 301 P = (hx-}-a^y)i and, for the equation of the evolute, after makina; c^ z= a^4-b^, — = m, -j-:=zn. The form of this equation shows that the evolute of the hy-perbola is coraposed of two branches of unlimited extent, andsymmetrical with respect to both axes of the hyperbola. Ithas two cusps situated on the transverse axis beyond the foci,and is convex at all points towards the transverse axis. ISd. Radius of curvature and evolute of the cycloid. du By squaring the value of —, which, for this curve, is ctoc __ |2r- (Art. 146), and differentiating, we find dy dx Substituting these values of -^-, ^yi ^^ ^^^^ general expres-sion for /), we have ^dy d^y __ 2rdy^ . dy r ^ dx dx^ y^ dx • • dx^ ~ t ^r^y r- .p=2V2ry. r Now, Fm = IN, and,from the right-angledtriangle PNG, wohave PN^VGNxNf;that is, FN^ the radius a a .^1 c )i 7 ^J^ NV ^ OmT^ p4 T \ D ^^ Vl/x / \ (4
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