200 L EULERI OPERA POSTHUMA. Astron.mech. 



67. Scliolioii !• Binae acquationes differentio-differcntiales in alias transformari possunt, ut 

 anguli (p sinus et cosinus elidantur. Uti enim Il.cosy — I.sin 9) .praebet 2dvd(p -^vddcp = {), ita 

 \.cos(f -\-\\.s\ncp suppeditat hanc aequationem 



quarum illa per v multiplicata et integrata statim praebet, ut vidimus, vvd(p = Cdt, qua aequabilis 

 arearum descriptio continetur. Deinde posterior per 2d^, prior vero per 2cd^ multiplicata in una 

 summa praebent 



2dvddv -I- 2vdvdq)''-+- ^vvdcpddq) = -^^(^1^^^^" d^ ^ -»- ^ 



quae integrata dat: . ,, v v. ..1 ik 



di>^-^i;pd(p^=Ddt^-^^l^^^dt\ 



ubi y^di^^^-^i^vdg)^) exprimit elementum spatii Bb tempusculo dt descriptl , iinde autem ob 



vvd^p"^ = altera aequatio integralis ante inventa elicitur. Juvabit autem has aequationes plu- 



ribus modis tractare, ut deinceps, cum hujusmodi aequationes magis complicatae occurrent, subsidia 



inde peti queant. Licet etiam has duas aequationes 



dp ^df ; 



Jie o^ 



■ ' ;'r:4)(|o ojjziijiraoin iiiu ^i, iw iC' y^ ^J;'- 6 ?.0'.i j\ 

 2dvd(p-^vdd(p = (i et ddv — vd(p'' -\-^-^^^^^ dt"" = ^ 



hoc modo resolvere: Multiplicetur prior per 2v^d(pj-ui'\iiibc2X\kT^%^ 



cujus integrale est v^^dcp'^^ EEdt"^, unde valor pro dt"^ in altera aequatione substitutus praebet 



^ dd^ - ,dy^-f- ^^ (^,-^B)..d,^ ^ ^^ 



Cum autem hic adhuc sit dt constans assumtum, ut ejus loco dg) tanquam constans introducatur, 

 multiplicctur per 2 dv, ut habeatur . i _ ^ •x 



2dvddv — 2vdvdy^-+- *^ (A-t-B)vvdi> ^^a^ q jj^^ » < u nia)iiB aiaj 



et loco 2di>ddv scribatur :;~;rvZ;-7""^ ?-' - (T^;;:;;/^ — »803«H-l V 



?. nra(«8o-)-i-S) tb A Iv) 



__^t- \ - — -T/ ., dk' —du j — /Tdu. 



df2 ££ y4d^ 



a(«80-)-i-S) tb 



et nunc elementum dcp est consl^ns. SlMuatiff^polTb ^f^^^v^» erit - = ^1^ et cdt^^— '4^ 



' * M t^l' ^ M' 



M" P ' ]\))uh<fiUu i\ io'^ oBub ofiils Q Jo oiDiJnGJzfloo fnmfinid oaol- oaollulozvi 0^19 acrf 



^d.^Vg^3Jg'_-*g<-^^^>^'^"^y?>.^;Qi^,onB ,(ii^IidGiir>7 mf.von r 



_ /•' 2dMddM 2dMdg)2 4fa(A-i-B)du , » „ .),.._,, l 



seu — - — -i ~ — — / d(c^ = 0, ; .1 



M* m3 ££m*' ^ ' 



vel 2dMddM-H2ac?aV--^^^^-^^^V=0, 

 cujus mtegrale est 'ioy.t^ «i xoai msnoiJjioikjqB Ja oiuzu «iaoiloloz zu[v 



