262 



L. EULERl OPERA POSTHUMA. 



Aslron.mech. 



dv qdcp sins cos w •\/(t 

 vv p COSf V 



C08^£ — cos'^ o 2 G (3 -t- A cos s) ' 



dv q sin 



'^d9>V{ 



COS^CJ 



2G (3 -*- A: coss) cos^cjv jsins 



|> seu 



O -t- K COS S) \ 



-If )' 



\ jsins/ , Gdf (3 -»- A coss) cos^o\ 



/ p \ ^ //"cos^f / 



vv p ' \ /fcos^t 



Per differentiationem autem obtinemus 



dp 3 (cc — aa) dcp sino coso sin^o (1 — 2ft coss -\~kk) 

 pdg — qdp 3 (cc — aa) d^s sinocoso sin^cj (coss — 2ft-HAftcoss) 



hincque concludimus 



ita ut sit 



d<p — ds = 



pp t' 



dv qds sin s 3 (cc — aa) drp sin o cos a sin^o (1 — kk) sin* s 

 vv p f^ 



gd^sins k{cc — aa)(l — 3 sin2<Jsin2cj)dg)(3-t-A:coss)cos^t)sins 

 p ^/■^cos^e 



3 {cG — aa) df sina coso sin^cj (1 — kk) sins (cc — ad) (1 — Ssin^o sin^cj) (3-i- Aicoss) d<p cos'cj 





ffk 2/7" cos* £ 



Cum igitur in his terminis minimis liceat ponere w = «, quae est inclinatio media, erit 



, (cc — aa)(3-*-kcoss)d<p 3 (cc — aa) (3-H*co»s) d?) sin^t sin^^o 3 (cc — aa) (1 —ii:*) d^s sin^t sin» sino coso 

 d(p aS= ^m 7r-:i 1- 



2/r 2/9^ 



ubi statuere licet dfp = ds = do» Tum vero habetur 



/r* 



P = 



fcos"^ £ (ce — aa) (1 — 3 sin^f gin^^o) (3 -*- **) 

 cos'^cj 2/" 



Wcos^e . cos^f (cc — oa)(l — 3 sin^E sin*o)(l — jfc*) 



qq = 2 »- 1 5- -" — -z 



' ■* cos-^w COS''^ ff^ 



ac praeterea 



j — 3 (cc — aa) (I -hAcoss) djo cos£ sin^o , — 3 (cc — aa) (1 -»- X;co8s)d)7 sinc cos£ sinocoso 

 dlp = , dw= ~ , 



eritque dcp = do -h- difj cos e , ac tandem pro tempore 



wd^jcoscj ppdq) cosci 



dt V^fgL 



COS£ €0S£ (1 -1-2 coss)' 



quae formulae omnes in terminis minimis sine difQcultate integrari possunt; postrema tantum formula 

 majorem solertiam postulat. Ponamus enim ad abbreviandum — — — = /i et evolutis productis 

 sinuum et cosinuum adipiscemur 



dtfj = — j^nd(p cose (1 -h Aicos* — cos2^ — l^ftcos [26 — s) — -l^ A: cos (2 <? -*-*)), 



dco = — -f /id^Dsine cose (sin2<7-*-^A; sin [26 — s) -4— |^/csin (2<7-»-5)), 



