. An elementary treatise on the differential calculus founded on the method of rates or fluxions. sin tp — 4 sin3 ^, we have x — a cos3 tpy — a sin3 tp (2) whenceand X^ = #3 cos2 ^,y\ = #t sin2 ^. XXXI. THE FOUR-CUSPED HYPOCYCLOID. 321 Adding, we have 2 % (3) the rectangular equation of the curve. This equation, whenfreed from radicals, will be found to be of the sixth degree. Example. 1. In the case of the four-cusped hypocycloid, express in terms of?/ the tangent of the inclination of the chord BP, Fig. 65, and provethat this chord is perpendicular to the tangent to the curve at thepoint P.
. An elementary treatise on the differential calculus founded on the method of rates or fluxions. sin tp — 4 sin3 ^, we have x — a cos3 tpy — a sin3 tp (2) whenceand X^ = #3 cos2 ^,y\ = #t sin2 ^. XXXI. THE FOUR-CUSPED HYPOCYCLOID. 321 Adding, we have 2 % (3) the rectangular equation of the curve. This equation, whenfreed from radicals, will be found to be of the sixth degree. Example. 1. In the case of the four-cusped hypocycloid, express in terms of?/ the tangent of the inclination of the chord BP, Fig. 65, and provethat this chord is perpendicular to the tangent to the curve at thepoint P. The Involute of the Circle, 301. The curve describedby any point in a straightline which rolls upon a curveis called an involute of thegiven curve. The curves de-scribed by different pointsof the rolling tangent usu-ally differ in shape, butwhen the given curve is acircle the involutes differonly in position. Let BP be one positionof the rolling tangent to a circle, and let the axis of x passthrough A, the position of the generating point when in con-tact with the circle, then BP= Fig. 66. By projecting the broken line OBP on the coordinate axes, wehave, OBP being a right angle, 322 CERTAIN HIGHER PLANE CURVES. [Art. 3OI. x — a cos ip + aip sin ip ) , , y = a simp — aip cos ip ) If the tangent roll back beyond the initial position, a cusp willbe formed at the point A, and the curve will consist of twosymmetrically situated infinite branches, as in Fig. 66. Example. 1. Show that the tangent to the involute of the circle is perpen-dicular to the rolling tangent, and find the maximum ordinate andabscissa in the first whorl. Max. ordinate when ip = 7t; Max. abscissa when tp = f n. The The transcendental curve defined by the equation y = -2(f+e is called the catenary, because it is the form assumed by achain, or perfectly flexible cord of uniform weight per linearunit, when suspended from two fixed points. The curve is evidently symmetrical with referen
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