AD TAVTOCHRONISMFM PRODFC. 151 



— V(yy-pp)zz.q, atque AN3s,AQ_r r,QN=: 

 V(s2— rr) — s. Erit radius ofculi ibi — j dy.dp. 

 hic yero zzzdz.dr. Deinde, elemefttum curuae 

 in illa —ydy.q-, in hac ~2^;s:r, vt ergo ele- 

 mentum ang. ATM fit ex illa curua zzzdp-.q, ex 

 hac ~ dr:s. Oportet vero efTe dp:qzzz — dr:s. 



$.15. His valoribus fubftitutis, proueniet haec 

 aequatio P bdp;qzz.Rdp — Sdr. Eft vero — dr:s 

 zzdp:q, vnde drzzz>-sdp;q , quo fubftituto aequa- 

 tio refultans Y b dp : q zzR d p -\- S s dp;q <\i- 

 uidi poterit per dp, quo fa&o, et multiplicato 

 per q. habebitur ifta aequatio ?bzzRq~\- Ss. Ex 

 qua haec fluit proprietas curuarum quaefitarum, 

 vt fumma R.PM-f-S. QN femper fit conftans fum- 

 tis tangentibus parallelis. Habentur ergo duae hae 

 aequationes ?bzzRq-\-Ss et s dp-\- qdrzzzo. Ex 

 qtiibus iundis problemati facile fatisfiet. 



§. 16. Applicemus haec ad axera AT. Dt?~ 

 cantur applicatae MX,NY; fitque AXzr .TjXMz:;, 

 et AY = ^YN = s. Erit PMr^^^ efi 

 QNzzszz^0^\ Atque ob tangentes PM,QN 

 inter fe paralielas erit dx:dyzzdv: — dz, feu fes 

 — dvdy.dx. Snperior vero aequatio P^zzrR^-f- 

 Ss transmutabitur in hanc TbzzR (xdx-\-ydy): 

 V(dx 2 -\-dy 2 )-\-S(vdv-\-zdz):V(dv 2 *-\-dz 2 )fub- 

 ftituatur loco dz,—dvdy:dx\ erit Vbzz R(xdx -\~ 

 ydy):V(dx 2 -\-dy 2 )-\-S(vdx-zdy):V(dx 2 -\-dy 2 ). 

 Vnde elicietur zzz vdx : dy-\-( R(xdx-\-y dy) — 

 ?b V ( dx 2 -\- <ty 2 ) ; ; S #f. Vocetur dx ; ir rr g • et 



(K{xdx 



