. An elementary course of infinitesimal calculus . Fig. 38. When SO is indefinitely diminished, the ratio of sin 8^ to 86tends to the limiting value unity, and 1 — cos BO, = 2 sin ^B0, 126 INFINITESIMAL CALCULUS. [CH. Ill is a small quantity of the second order. Hence we maytan PP0 = p^=-g^ +a (1), where o- is a quantity whose limiting value is zero. Henceultimately, when P coincides with P, we have, if <f) denotesthe angle which the tangent to the curve at P, drawn onthe side of ff increasing, makes with the positive direction ofthe radius vector, jo tan- Ex. \. In the circle r = 2a& (
. An elementary course of infinitesimal calculus . Fig. 38. When SO is indefinitely diminished, the ratio of sin 8^ to 86tends to the limiting value unity, and 1 — cos BO, = 2 sin ^B0, 126 INFINITESIMAL CALCULUS. [CH. Ill is a small quantity of the second order. Hence we maytan PP0 = p^=-g^ +a (1), where o- is a quantity whose limiting value is zero. Henceultimately, when P coincides with P, we have, if <f) denotesthe angle which the tangent to the curve at P, drawn onthe side of ff increasing, makes with the positive direction ofthe radius vector, jo tan- Ex. \. In the circle r = 2a& (4) we have log r = log 2a + log sin 6, and therefore —=„ = cot 6, r dd whence cot<^ = cot5, or = 0 (5).. oFig. 39. * The argument, which is an application of a principle stated in Art. 30,may be amplified as follows. We have, exactly, and the limiting value of this is evidently rdBjdr. 55] APPLICATIONS OF THE DERIVED FUNCTION. 127 Ex. 2. When the radius vector of a curve is a maximum ora minimum, it is in general normal to the curve. For if drldO = 0, we have cot = 0, or <^ = Jir. EXAMPLES. XVI. 1 Prove that the condition that the tangent to a curveshould pass through the origin is X dx Prove that a pair of straight lines can he drawn through theorigin, each of which touches all the curves obtained by giving cdifferent values in the equation y = c cosh - . 2. Prove that the perpendicular drawn from the foot of theordinate to the tangent of a curve is y- Hi))- Hence shew that in the catenary y = c cosh xjc this perpen-dicular is constant. 3. Prove that the perpendicular from the origin on thetangent is (»-|)M-(l)}* Verify that in the circle this perpendicular is constant, an
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