58 



L. EULERl OPERA POSTHUMA. 



Mechanica. 

 in plagani 



II. Mom. circa OB = 



2d£ 

 d(2 



CA 



in. Mom. circa OC = — 



AB 



h- yz {'Udd^ -i- !Ddd(E-4- md^) 



— xy {(^dd^ -4- ^dd® -t- ^dd^) 



- zz {%dd(^ -+- ^dd^ -+- md^) 1 

 \—xx{(^dd^-+-^dd^-^^dd®)\ 



- xz {md% -f- ^dd^-i-md®) 

 ■ xz ((5(ld(5-4-gddg-+-3dd3) 



-£cz {^dd(^-h-(^dd^-*-^dd^) 



— jr (5tdde -+- t>(idt5 -»- ®dd^) 

 -xx^^dd^-i-^dd^^-i-^dd^By 



— jT (5l(id33 -H ^ddd -H ®(idJg) 



- a;j {^dd^ -*- (SdciCS -+- ^d(?^) 

 ^ xy {md% -^ '^dd^-i-(§^dd(3) 



67. Positio autem trium axium mobilium OA , OBy OC in spatio absoluto commodissime cogno- 

 scitur ex his tribus angulis 



.,3aA-^r. aA = p et aAB = q. 



ex quibus fit, ut supra vidimus, , , 



>b (nutoai bc yupo. » 



3t = cos/), 93 = sin/) cosg, 6 = sin/)sin7, 



5D = sinpcosr, @ii= — sinr/ sinr ~ cosp cos^cosr, ^ = -j-cos^ sinr — cos/j sin^ cosr, 



® = sinp sinr, j^=:4- sing cosr — cos/) cos^ srnr, 3 = — cosgcosr — cos/) sin^sinr. 



Ex his ergo tribus angulis non S6luai ratio aovem quantitatum 91, 93, (S etc. sed etiam earum dif- 

 ferentialium definiri conveniet. 



68. Erit autem differentialibus sumendis 

 d%= — dpsinp, d^ = dpcospcosq — (^dq, d(^=dpcospsinq-*-^dq, 

 d2)=c(/)cos/)cosr — (S(?r, ^©^(^'/jsin/^cosgcosr — ^dq — ^dr^ d^=-i-dps\npsinqcosr-¥-Q,dq — ^dr, 

 d(B=dpcosps\nr-^Mr, d^=dps\npcosqs\nr^^dq-\-(§,dr^ d^=-\-dps\nps\nqs\nr-^^dq-+-^dr. 



69. Hinc autem porro eliciuntur istae formulae 



3t=(S3--s§, ^=m-^3, 



6=3:5^— (s@, 



2) = ©^ — 933, ^ = 9(3 — (5®, e = 33® — 21^, 

 (§ = 93^^(5®, ^^g2)_9(^, 3 = 5t(E — 93T), 



harumque ope sequentes : \ \^ /,, 



^d(S, -i-2)(Z^ -i-®el3[^ = (Z/)isitt7-i-(irsin/)cosg, 

 ^d% -i-%d^ -t-3cf(^ — — dpsmq — cZrsinpcos^, 

 d^d(B-t-d^d)^-i-d(Bd^= (dpcosq — (ir sinpsin^) ((Zr cos/) 

 ^dSd -t-^d(i -i-^d^ =^ dpcosq -^drslnpsmq. 



)faom Biii ottnftup 



dq), 



