De curvis, quarum rectipcatto per quadraiuras mensuralur. 449 



vero cx punctis M et &d rectam Cj4 pro axe assumtam demittantur pcrpendicula MP et OS^ 

 itemque ex M in rcctam MT axi /4C parallclam, sintque coordinatae ortbogonaies curvae descriptae 

 CP = x et PM = y. 



26. Cum jam sit angulus j4CQ = cp et CO = a±ry ubi signum superius pro curvis epicy- 

 cloidalibus, infcrius vero pro hypocycloidaiibus valet, erit C S = {a±r) cos (f ct OS=(adt.r)^\n(p. 

 Deinde ob sc[i^.COS=W—(p et COR = u(p, crit an^. MOT=(u-i- i) (p — dO° pro cpicycloi- 

 dalibus (Figg:.50, 51), at pro hypocycloidalibus (Figg.52,53), ob COS=dO°—(p et COR=iS0°^u(p, « 

 erit angi>/OT=90° — (a — 1) 9), unde ex triangulo OMT ad T rcctangulo, ob latus OM = /nr, 

 obtincbimus pro utroque casu 



curvarum hypocycloidalium Fig. 52ct53: 

 MT = jurcos (u — t) cp, 

 OT = /ursin (u — 1) ^, 

 ergo CP ={a — r)cosf/-*-^rcos(« — i)(p=x^ 

 PM= {a — r^sin^r-f-Mr sin (« — i)(p=y. 



curvarum epicycloidaiium Fig. 50 et 51 : 

 MT= — fircos{a-\-i)(p^ 

 OT = -h- /ursin {pc-t- 1) ^, 

 ergo CP = {a-i-r)cosq) — iurcos{u-^i)(p=x, 

 PM= («-♦-/•) sin q) — /urs\n{u-i-i)(p=y. 



Consequentcr pro utroque casu conjunctim 



CP = x = {a ±r) cos 9? q= ^r cos (1 zt— ) (p, 



PM = y = {a±r) sin (p^ /ir sin (1 ± ) (p, 



27. Ilinc crgo vidcmus totum discrimen inter has curvas epicycloidalcs et hypocycloidales 

 tantum in signo quantitatis r esse situm, ita ut omnes his expressionibus pro coordinatis CP = x 

 et PM = y possimus complecti 



x = {a-^r) cos (p — //r cos (1 h — ) cp, 



y = (a -H r) sin y — /^r sin (1 -*- y ) 9P, 

 quae proprie ad cpicycloidales pcrtinent, scd sumta quantitate r negativa simul ad hypocycloidales 

 extenduntur. Diffcrentiando crgo habcbimus 



dx = — (a-*- r) d(p fsin (p — /li sin (1 -*- — ) (p\ 



dy = -\- {a -\- r) d(p icos (p — //cos(l-t- ) fp\ 

 undc elementum arcus hujus curvae "|/((Za?^-i- dj^) = rf* rcpcritur 



ds = {a-\-r) d(py{i -^ /1/1 — 2/< cos (p), 



et radius osculi in M ita erit cxprcssus: 



a i 

 (a-i-r)(l-i-A«At — 2/< cos — f)^ 



a a 



Hfi — /t(2-*- )cos--«) 

 r r 



28. Quaccunque igitur hujusmodi curva dcscripta dabitur cllipsis, in qua arcui curvae DM 

 arcus aequalis assignari potcrit. Sit (Fig. 5h) adbe hacc ellipsis, cjusque axes orthogonales ab ct de; , 

 vocetur semiaxis minor ca = cb = c ct scmiaxis major cd = ce = mc, sumtacjuo super illo a conJro 

 c abscissa cp = z, crit applicata pm = mY{cc — zz) ct arcus ellipfitus dm= /'dzV{i -\- — — )• 

 Statuatur z = c sin 6^, eritque hic arcus 



L. Euleri Op. po$th«ma T. 1. 57 



