224 L. EULERI OPERA POSTHUMA. ^ Astron.mch. 



ddX^^^di^cos^, ddY= -^ dt^ cos^y, ddZ==-^ dt^cos &, 



vv vv vv 



ddX-*-ddx=^=^dt^cos^, ddY-^-ddy = ^^dtHosrj, ddZ-\- ddz = ^^ dt^cosf^. 



Hic quantitates X, F, Z referuntur ad motum absolutum corporis J, a.t x, y-y z nd raotum re- 

 spectivum, quo corpus B c\ A spectatum moveri cernitur. Pro hoc ergo coUigimus: 



< ,7 — 2o(^-+-B)df2co8S — 2sf (4-+- B)a!dt2 



I. ddx = ^^ = z f 



VV V' ^ 



„ jj — 2sf(J-i-B)df2cosi? —^g{A-^B)ydt^ 



II. aay =^ = 5 , 



" VV i^" 



111 jj —^g(A-i-B)dt^cos9 —^g(A-t-B)zdt^ 

 111. aaZ =: = 5 i 



vv v* 



ex quibus cum aequationc xx -t- yy -i- zz = vv omncs quantitates cc, y^ z et c ad tempus t deter- 

 minari oportet. Inde autem primo has aequationcs integrabiles deducimus 



yddx — xddy = , zddy — yddz = , xddz — zddx = , 

 quae integratae dant 



ydx — xdy = Edt^ zdy — ydz = Fdty xdz • — zdx = Gdt. 



Quare cum sit F(ydx — xdy) = E(zdy — ydz), per yy dividendo nanciscemur — =— — i- Const. 



Similique modo ob ^(zefy — ydz) = F{xdz — zdac), pcr zz dividendo adipiscimur — = — ^-^ -f- Const. 



cx quibus conjunctim deducimus Fx -¥- Gy -i- Ez = (i , qua aequatione motus corporis B ex A spec- 

 tatus in eodem plano fieri indicatur. 



Porro si primam per 2dXy secundam per 2dy et tertiam per 2di multiplicemus , ob 

 xdx H- ydy -\- zdz = vdv, summa erit 



2dxddx -f- 2dyddy -^2dzddz = ''*^^'^^^^^^ dt\ 



cujus integrale ob dt constans dat 



dx'^dy'^-^dz'=Ddt^-^ ^^^'''*-"^^'% 



V 



ex qua ope aequationum 



Fx -t- Gy -+- Ez = f xx-^yy-^zz = vCy ydx — xdy = Edty zdy — ydz = Fdt, 



xdz — zdx = Gdt 



eadem solutio deducitur, quam jam supra dedimus. Denique inventis variabilibus cc, j, z motum 

 respectivum spectantibus , ex iis pro motu absoluto corporis A colliguntur coordinatae X, F, Z per 

 has aequationes 



A-*-B ' A-t-B A-^B 1 



