230 L. EULERl OPERA POSTHUMA. * Astrm.mech. 



cum quibus aequationem vel differentjo-differentialem ddv, vel integralem inde natam combinari 

 oportet. 



13'i'. Problema. (Fig-. 172). Si corpus M, quod ut quiescens spectatur, habeat momenta 

 inertiae principalia respectu axium X^ et JB acqualia, idque sive quiescat, sive circa 

 axem tertium JC gyretur, defmire ejus respectu motum alterius corporis sphaerici N. 



Solutio. Retentis omnibus denomlnationibus, quas in problemate praecedente constituimus , ob 

 bb = aa, aequatio nostra integralis erit 



da.2_^ ^^2_^ d^^ 2)^,2 _^ ^^ ^j^_^_pf, df. (^l ^ gaayc_^ 3aa(..-^y-3ccz. y 



Praeterea vero habebimus has duas 



yddx — xddy = et xddz — zddx = —^ ^ ^ — 



existente xx -\- yy -\- zz = vv , quarum illa integrata praebet jdfcc — xdy=Edl. Statuamus praeterea 

 z4y — ydz = qdt et xdz — zdx = rdt, eritque , !■ 



-, Gg{cc — aa)(M-*-N)yzdt , —egicc — aa){M-t-N)xzdU ^ 



aq = ^;t et dr = tr jj ' ^5 ' 



ita ut sit xdq-t-ydr = 0. Tum vero ex illis tribus formulis colligimus Ez -t- qx -¥- ry = , atque 

 insuper {EE -^qq-h- rr) dt^=vv {dx^ h- dy^ -\~ dz^) — vvdv"^ , hincque 



Ac._H af^ d,.= ^,._^(^^-^.»H-rr,d,n = -w> 



Ex aequationibus Ez -\- qx -¥- ry = et xx -\- yy -^ zz = vv concludimus 



— Eqz-^rV{{qq-*-rr)vv — {EE-t-qq-^rr)zz) — Erz-^qV{{qq-*~rT)vv — {EE-*-qq-*-rr)zx) 



x = '■ f y= • 



•Deinde Tero habemus dz=— — qy—rx ^ ((jcoque 



^ V vv ^ 



, zdv dt _^,,, . ,„„ . , M. j ^ * n// {EE-*-qq-t-rr)tt\ 



dz=---^~^y{{qq-+-rr)vv — {EE-h-qq-i-rr)zz) et d- - = -^y[qq -+- rr ^^ j. 



Aequatio autem prima hinc reducitur ad hanc formam 



*^ = Ddt' - '^^-"^^J"^'"' -H 43 {M -^N)dt' (i -H ("'-'"H7-8") '). 

 Ponamus z= uv, q = s cos co et r = 5 sin «, eritque 



dy^= Ddf^- ^^^-^;;^''' -V- hg [M-^N) dt^ (1 -h (-<--"«^<^^-'^»-'«) ) 



atque da = — y{ss — [EE -*- ss) uu) ; tiim vero f;^ 



4 



vv 



■ tl 



— £ coso -♦- sin o "/(ss — (££ -+- ss) uu) — £ sin — coso V{ss — {EE -*- ss) uu) 



x= ^ -'V, y = ^^ !^ — --V, 



unde differentiaha dq et dr supra defioita dant 



