. Differential and integral calculus, an introductory course for colleges and engineering schools. ,/3 = + cos 0 • Dar - r sin , by formulae III, Art. 101, and I, Art. 108, 1 HenceWhence Dsx = cos 0, Day = sin 6, Da6 = a = Dsa = cos 9 — sin 6 • Dar — cos 6 = —sin 6 Dar,Dap = sin 0 + cos 6 • Dar — sin 6 = cos 6 Dar. Dsa aH tan 0 1 But DajS is the slope of the tangent to the evolute at C, and - tan 0 is the slope of the normal to the given curve at P. These two lineshave therefore the same slope, and since they have the point C incommon, they coincide throughout, which proves the
. Differential and integral calculus, an introductory course for colleges and engineering schools. ,/3 = + cos 0 • Dar - r sin , by formulae III, Art. 101, and I, Art. 108, 1 HenceWhence Dsx = cos 0, Day = sin 6, Da6 = a = Dsa = cos 9 — sin 6 • Dar — cos 6 = —sin 6 Dar,Dap = sin 0 + cos 6 • Dar — sin 6 = cos 6 Dar. Dsa aH tan 0 1 But DajS is the slope of the tangent to the evolute at C, and - tan 0 is the slope of the normal to the given curve at P. These two lineshave therefore the same slope, and since they have the point C incommon, they coincide throughout, which proves the theorem. Theorem 2. The difference in length of any two radii of curva-ture is equal to the length of the arc of the evolute included by them. Proof. Let C\ and C2 be the centers of curvature of P\ and P2respectively, and let ri and r2 be the corresponding radii of curva-ture. We wish to prove that r2 — n = arc has just been proved that Dsa = — sin 6Dsr, Dsp = cos and adding these equations, we have(Dsa)> + (ZW = (Dsr)\ 160 DIFFERENTIAL CALCULUS §111. Iir? Now if s represent the length of an arc of the evolute, it follows from formula II, Art. 101,that (D8ay + (D8(3y = (D8s)\Hence D8s = D8r, Dss - D8r = 0,x D8(s-r) = 0. Now a function whose deriv-ative is 0 is a s — r = k. Let A be the point of the evolute from which s is measured, andlet ACi = Si and AC2 = s2. Then «i — rx = h, and s2 — r2 = fc, whencesi — ri = s2 — r2, and r2 — n = s2 — si = arc CiC2. Q. E. D. By aid of these theorems may be devised a simple mechanismfor tracing the involute when the evolute is given. The relation7*2 = ri + arc CiC2 is one which holds for all positions of Px andP2. Thus (see figure following) (r) r2 = ri + arc CiC2, r3 = rx + arc CiC3, r4 = n + arc CiC4, etc. On the evolute ACi . . C5B as base construct a right cylinder(of wood or metal) and let a cord be fastened at some point this cord be wrapped fora portio
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1912
