39<^ DE STATV AEQyiLIBRII ET MOTV 



A M rz j- pondus erit ~ , ideoqne pondus tkmenti 

 Mm erit — -^ , cui aequari debet vis illa P^x, fit 

 autem breuitatis gratia -1-=:^, vt fiat PzzijS, atque 

 ob ^ zn o habebimus p cr. p fin. Cf) , 9 = — P coK (p 

 ideoque p d s :=^ ^ d x et q d s zz - p d y. Qiio.uam 

 hoc problema ad primum cafum pertinet , habebi- 

 mus fequentes fbrmulas : 



I. dT-^dx et 11. -i-Td(p-^dj, 



Ex priore fit T 3 p a' 4- C , ideoque hinc pro curua 

 coUigimus (i x d (p ^Cd^-pdy 

 ad hanc aequationem reloluendam ponamu^ ftatim dy 

 z:udx,fLQt(\ucds::r.dxy(i-^uu), hinc C\n.(p — ^^-^ 



et cof. Cl) — : vnde elicitur d(P z^ "^, quo 



valore fubftituto aequatio noftra erit zt-Jjl^ p x -h c j 

 — audx; ideoque ^^l^--f:AJ! zz^^X^^l^, 

 vnde integrando confequimur Log. ip;r4-C)- L^ ^' "^ :."— ^ 

 + L.D, feu (3.' + C=D^ii\ vnde uz^-^^-^^^ 

 rz^, hincque ^ y =i: ^^^^ ^ ^J^-g^, , quie eft ae- 

 quatio differerentialis inter coordi latas x et y pro 

 catenaria , cuius conftrudio pcndet vti conftat a 

 logarithmis, fiquidem hinc fit ^^-L P ^"^^^-f ^ ^^^-^^^ 

 praeterea vero notafte iuuabit , hinc fore 



/f c — (^x-^C)dx 



" -* — V (( ^3 X -h C)^ — D D) 



ita vt fit Dds — ipx-hQdy , inde vero intcgrando 

 colHgimus ^ s — y {{px + Cy -DI)) + E vbi (3/ 

 dcnotat ipfum pondas arcus A M.zz s, 



XXIV, 



