(;> )

sion of thc Arc H M, about the Axis C A is equal to — xP Cx HM

-RAxW Z,whenthepointZ isnexttoQor— X PCxHM+ RAxWZ,

r

when the Point W is next to it.

Those that will thinkit worth theil' while tobeflow somc little Pains tofindthe Demonilration of this, may solve the following Problem.

Any tno Conic ScSlions being givex, forming a Lunula by their Interfeflion, anda l{igbt Line being given by Position, about vehich, as an Axis, thisLunula is imagined to turn, to find the Solide generated by the Converfionof any of its Parts, cut off by Lines Perpendicula) to that Axis, or Paral-lel to it, er makjng any given Angle vtith it, as also the Surfaccs made bythat Converfion.

IX. Suppose D P V, to be half of an exterior Epicycloid, V B its Axis, rhe o Uill { s jture >V the vertex, VLB half the generant Circle, E ks Center; DB the Base,»/ -C its Center: Bisect thc Arc of the SemicircleVB in L, and on the Center C *hc Epicycloid,tythro’ L, draw a Circle entring the Epicycloid in P: then I sey the Curvilh

G fc* , F ig, 2?*

near Triangle VLP will be — B E <j in ; that is, the Square of the Se~

midkmeter of the Generant Circle, will be to die Curvilinear Triangle VLP,as C B the Semidiameter of the Baie to C E : which C E in the exterior Epi-cy.cloid is the Summ of thc Semidiaraeters of die Base and Generant, but inthe Interior Epicycloid D pu, ’tis the differenee of the seid Semidiameters.

C O \ 0 L. I.] In the Interior Epicycloid, if C E is C B, the Epicy-cloid then degenerating into a Rigkt Line, the Qyadraturc of the Trianglel p u, will be in efieef the seme with the Qyadrature of Hippocrates Chius.

C O R^O L. II.] If the Semidkmeter of the Base is lupposed infinite, theEpicycloid then being the Common Cycloid, the Area of the seid Triangle•.vili be equal to the Square ofthe Radius of the Generant; and Id it falls in withthat Theorem which Lalovera found and calls Mirabile.

The general Propofition from whence-I deduced the-a bove seid Quadrature,is this ; vi\.. the Segments of the Generant Circle are to the Correjpon-dent Segments of the Epicycloid , asGB, to t C £ -f C R ForExample. Suppose F m G, the Position ofpart of the Generant, when thepoint F of the exterior Epicycloid was designed, then the Segment E»G».is to thc Segment DF »G, as C Ero iCE + CB. And cansequently thewhole Epicycloid to the whole Generant in the fame Proportion , which. isthe only Case demonstrated by M. de la, Hire.

It follows alio, that in the vulgär Cycloid, ks Segments are triple of the ■

Correlpondent Sectors of the. Genannt, which was first Ihewn by Dr. Wallw.

X; Arca.;.