26a L. EULERl OPERA POSTHUMA. 



quibus comparalis cum ante assumtis erit L = M-t-N et 



Aiiron. mechj 



1« I 



x = 



Sx(4aa-*-cc) lox (aaxx -t~ aayy-t-eetz) 



21»* 2f' 



,;. Sy (4 aa -+- cc) 15y (aaxx -+- aayy -4- cczz) 



575 "~* qTTT ' 



^i'^ 



2i'' 



z=- 



32(2aa-«-3cc) ISz (oaaraJH- aayy-e- cczz) 



2f^ 



2w' 



hinc ob xdx -+- yd/ -4- zdz = vdv^ erit 



_,, ,rj v^ 3(4aa-i-cc) (arda; -f-yrfy) 3 (2aa>i-3cc) zdz 15A' (aaxx-4-aoyy-»- eeix) 



-' 'iv^ 2v^ 'iv* 



3 (4 aa -♦- cc) df ^{aa — cc)zdz ISaadt^ 15 (aa — cc) zzdf 



Ergo AXdr -^ Ydy h- Zdz) = ^!^ - ?i?^% hincque 



Cum nunc ex § praecedente sit yddx — a?dd[y = 0, erit 



ydx — xdy = — wd(f>CQS(o = — EdlVkgL et vvd(p^~— ^ — , hincquc 



j 5 r r j^2 /rv * ec — aa 3(ce — aa)zz 

 d^^=kgLde{L-^--^-^ — 



EE 



2f3 



2v5 



i'V C08* o 



) •« 



- , v*dw^coa^c} ,r\ 1 cc — aa 3(cc — aa)zz EE v 



cit'»^— -ii:^^ — (D-i »--^-i a 5 2"), seu 



EE 



2^» 



2t>s 



\ £(fc' , ,//rv 1 cc — aa 3 (cc — aa) sin^o gin^tj EE v 



— — dg) cos fij y (D H H -3-5 ^^ 5-5 j-) 



atque 2EdtygL = {>vd<pcos(a. Deinde vero habemus 



P^^d^p^cosVsin^w^ 12^L(aa — cc^df^^/^^^t^P^ et 



v'd^^ sinVsin^to = 12^1 (aa — cc) d«» /^'^''7^'°" , 

 quibus additis fit 



c*dqp' sin*a> = i2 gL (aa — cc) dt^ J — 



Cum porro sit v^d<p* cos^(o = kgLEEdt^^^ erit differentiando 



2f*<iy*ci(»sin wcos w -«-sin^wd. (f^d^?'^)^: i2gL (aa — cc) dt^ - '°°''°'^"° 

 et — 2c* dqP^dco sin w cos w -♦- cos*a) . d (^-^ d^p*) = 0, 

 unde concluditur 



