228 L. EULERl OPERA POSTHUMA. Astron,meck. 



corporis M momenta principalia fuerint inter se aequalia, difficultates illas maximam partem evanes- 

 cere. Statuamus erg-o momenta inertiae respectu axium X^ et JB inter se aequalia, seu bb = aa^ 

 atque evidens est in hac hypothesi perinde esse, sive corpus M quiescat, sive ei motus gyratorius 

 quicunque circa axem tertium JC tribuatur, quoniam omuia momenta respectu axium in plano JJB, 

 . quod tanquam quiescens spectamus, sumtorum, sunt inter se aequalia. Pro hoc ergo casu motum 

 respectivum alterius corporis sphaerici N investigemus. 



133. Scliolion 2* Interim tamen conatus exposuisse juvabit, qui forte aliquando ad solutio- 

 nem producere valeant. Faciamus statim has substitutiones 



x = pCy ydx — xdy = Idt, 2g {aa — hb){M-^N) = A, 

 y = qVy zdy — ydz = mdt^ 2^(66 — cc) (M -*- N) = B ^ 

 z = rv, xdz — zdx = ndt, 2g {cc — aa) {M-t- N) = C, 



ex quibus pro novis litteris concludimus has relationes 



pp -V- qq -\- rr = i , Iz -\- nix -\- ny = ^ , seu Ir -\- mp -\- nq = (i , 



tum A-\-B -\-C=0, item Jcc -h Baa -+- Cbb = 0. Porro ob yddx — xddy = dldt, habebimus 



j, —SApqdt , —SBqrdt , —SCprdt 



dl = P—i dm = 3^> dn = f— • 



y» J>* ^i 



Deinde cum sit ydx — xdy = vv {qdp — pdq) nanciscimur 1 



7 , Idt , , mdt . , ndt 



qdp — pdq=—i rdq — qdr = — > pdr — rdp = — > 



undc fit ^ — ^^^ = qqdp — pqdq — prdr -\- rrdp = dp , quia est — qdq — rdr = pdp et 

 pp -\- qq-\- rr= i. Sicque erit 



(?g-nr)dl^ ^ (mr-lp)dt^ ^^ ^ {np - mg) dt ^ 



* VV * Vi> vv ^ f 



atque hinc porro colligitur rdl-\-pdm -\- qdn = 0, mdp -*- ndq -t- Idr = 0; tum vero etiam _ 



ccrdl -\- aapdm -\- bbqdn = ,* ideoque Cpdm = Bqdn, Crdl = Aqdn, Brdl = Apdm, 



rdl pdm qdn — Spqrdt . .• 



Ex assumtis autem aequationibus obtinemus 



dt^ {II -\-mm-\- nn) = vv {dx^-\- dy'^-^- dz^) — vvdv^, 

 ita ut nostra aequatio integralis futura sit 



^^_^{U-,-mm-^nn)dt^ ^ j)^^2^2g {M -\- N) dt^f- -\- «« -^ ^^^ -^ '' 3 {aapp -H hh^qq -^ ccrr) \ 



in qua, quia quantitates ll-\-mm-\-nn et aapp-\-bhqq-\-ccrr , investigemus per formulas 



superiores earum differentialia. Reperiemus ergo 



