S63. 



AH; ^XSSAY ON CTCLOIDAL CURVES. 



EAC=ACR, RA=RC=:RB=RP, ABrJSC, and drawing 



the perpendkulars CT, TD, TE, and MF, RM=;RT, AM 

 =BT, AF=EC, FC=AE, and FM=BD. Let SCzza, 

 FCizr, FM=y, CM=s, CT=« ; then AB : AC : : AC : 

 at:: AT : AE, whence ATzz{axx)i, and in the sameman- 

 "^"^ ^^^i'^yyH j andCT beingamean proportional between 

 AT and TB, u^—{a'x'y-')L, and u'~a''x''y\ Butbyprop. 7, 

 3u'=a-— s% therefore 27a'j?'j/'=:(a«_s»j'z=(a"— x'— 

 y')' ; whence the equation may be had at length by involu- 

 tion. The same result may be obtained by Dr. Wariilg's 

 method of reduction, from (a.r.x)'+{nyij)'<—a. 



CoROLiABY. Since tlic portion of the tangent AB inter- 

 •ccptcd between the perpendiculars AC, BC is a constant 

 quantity, this hypocycloid may in tliat sense be called an 

 equitangcntial curve ; and the rectangular corner of a pas- 

 sage must be rounded ofT into the form of this curve, in 

 order to admit a beam of a given length to be carried 

 round it. 



Proposition XII. Problem. To inves- 

 tigate those cases in which the geiieral pro- 

 position.s either fail or require peculiar mo- 

 difications. 



Case 1. (Plate 8. Fig. es.) If the generating circle be con- 

 «idered as infinitely small, orthe basis as infinitely large, so as 

 to become a straight line, the epicycloid will become a com- 

 mon cycloid, and the ratio of CP to CK in prop. 3, cor. 2, 

 becoming that of equality, the length of the arc SM will 

 be four times the versed sine of half PM, and VM twice 

 tlie chord RM or VX, therefore the square of the arc VM 

 is always as the absciss VZ. The evolute is an equal cy- 

 cloid, and the circles in prop. 6 being as 1 to 4, the area 

 of the cycloid is to that of its generating circle as a to'l. 

 The properties of the cycloid as a'n isochronous and as a 

 brachistochronous curve belong to mechanics, and it is de- 

 monstrated by writers on opfics that its caustic is composed 

 of two cycloids. 



Case 2. '(Plate 8. Fig. 69.) If the concentrating circle be 

 supposed to becomeinfinite while the base remains finite, the 

 epicycloid will become the involute of a circle ; and the 

 fluxion of the curve being always, by prop. 3, cor. 1, to that 

 of PM as PM to CI', its )eng.th SM will be a third propor- 



tional to IP and PM. Call CP, a, and PM, .r, tficn the 



fluxion of SM is '— ; but the rectangle contained by 



half PM and the fluxion of SM is the fluxion of the area 



PSM, or PSMz: /_! = —. The epitrochoid described 

 t/ 2tt 6 a 



by the point C of the generasin.e; plane will be the'spiral of 



Archimedes, since CN is always equal to PM=PS:i:QV , 



and; since the angular motion of CN and PM are also 



equal, the area CON = PSM=z— . Instead of the ellipsis 



of prop. 3, let PX be a parabola, of which IP is the parame- 

 ter, and continuing NM to X, the arc PX will be equal to 

 CON. For making LH=CP, it is well known that tlie 

 fluxion of PX varies as XH, or as PN, which represents 

 the fluxion of COX. For the curvature, PY, in prop. 4, 

 becomcs^zzCP, and the radius is a third proportional to 

 NZ and NP. 



Ca.'x: 3. Supposing now the generating circle to become 

 again finite, but to have its concavity turned towards the 

 basis, the same cunre will be described as would be de- 

 scribed by the rotation of a third circle on the same basis in 

 a contrary direction, and equal in diameter to the difTcr- 

 ence of those of the two first circles. 



Case 4. If the circles be of the same si^e, with their 

 concavities turned the same way, no curve can be described; 

 but if tlie geneiating circle be still further lessened, a hypo- 

 cycloid will be produced, of the same figure as that which. 

 wouM be described by a third circle equal in diameter to the 

 diflfcrence of the two first. All the general propositions are 

 equally applicable to bypocycloids with other epicycloids, 

 as might easily have been understood from an inspection of 

 the figuresj if there had been room for a double series. 



Case 5. (Plate s^Fig. 70.) If the diameter of the generat- 

 ing circle be half that of the busts, the hypocycloid will be- 

 come a right line, and the hypotrochoid an ellipsis. For since 

 the angle PKM=:2 PCS, PCM, being half PKM, coincides 

 with PCS, and M is always in CS. Let GNL be the de- 

 scribing circle of the hypotrochoid, and join GNO; then NL 

 is parallel, and ON perpendicular, to SC, and ON=:HL, 

 which is always to GO as CL to CG ; therefore AN is an 

 ellipsis : and the centre K will evidently describe a cird«. 



