. 266 



L. EULERI OPERA POSTHUMA. 



Aslron . mechl 



OQ = OP -i- X cos ^ , QM ==:PL-t-x sia ^ 



OR =-- OQ -i- r cos rj , RN= QM-^ y sin v 



0P= 0R-1-ZCOS&, PL = RN-i-zsm& 



hincque colligimus 



X cos ^ H- y cos tj -Y- z cos & = ct X siti ^ -i-y sm 1] -h- z s\ti& = 

 ac porro 



X sin (^ — &)-^y sin (jy — »9^) = 0, £c sin (^ — ;?) -h z sin («9^ — ^) = 0, 



y sin (j; — ^) -i- z sin («9^ — ^) = 0, 



ideoque oj : y : z = sin (?? — 19^) : sin (»9^ — ^) : sin (^ — ??) , 



unde relatio inter distantias et angulos ita commodissime exhibetur, ut sit 



x = csin(t; — »9^), j = csin(»9^ — ^), z = p sin (^ — ^), 



ubi V denotat diametrum circuli triangulo LMN circumscripti. Si jam massae corporum litterij 

 cognominibus L^ M^ N exprimantur, corpus L a reliquis sollicitatur 



^_ . XiWcosS LN coi& 



sec. OP VI = — 



XX zz 



et sec. PL vi = > 



XX zz 



corpus vero M a reliquis soUicitatur 



:. 00 VI = — ^ et sec. QM vi = 



sec. 



\fU 



et corpus iV a reliquis sollicitatur 



LN co$& MNcosii 



sec. OR vi = 



yy 



i/y 



■miT • iiVsin^ AfiVsinif 



et sec.RN\i = , 



zz yy 



dd . PL = 2 ade^ (^^ - ^)^^ , 



«^ \ xx zz / 



dd.QM= 2gde (^ - ^V 



dd.M 



0«/7#2 /^sjn^ J^»P? 



2yrf<2 /: 



) 



unde sequentes aequationes adipiscimur 



'^ \ XX zz / 



dd.0R = 2sdt'(^-^). 

 ex quibus colligimus sequentes 



,, ^ rk J.2/ (L-t-M)cosi Ncosii iVcos^N u . y. -^ j^i / (Z-i-iW)sin5 iVsin-j .Vsin^\,J 



dd. xcosC = 2 qdt^i — ^ — --i '-i )) dd.xsinC='2qdt^ — ^^ — =h '-i )r 



^•'Vs XX yy zz J '^ \ xx yy %z J 



,, o j^9 / (itfH-iV)cos;/ Zcos^ £cosS\ ,, . -»1.5/ (M-^N)sinti ZsinJ? Xsin^N 



dd.ycosti = 2gdt^{ — ^^ — -'h 1 ]i dd .y sinrj = 2qdt^ [ — ^ — '-\ 1-- — h 



J ' <r \ yy ZZ XX J -/ ' O \^ yy zz XX J 



,, n rt .iaC (L-i-N)cosd^ Mcos^ Mcos?!\ j, . n - ,5/ (I-f-iY^sin^ .MsinJ -IfsinA 



rf^.z cos »9^= 2 flrt/r( — ■ ' — -H ^H -]t dd,zsin& = 2qdr{^^ y--— -\ i! 



''\ zz XX yy J "^ \ zz xx yy J 



quae porro transformantur in has 



