2 32 



Si £'"3 .te'—'" . 



dz 2!lZ 



dz £z"' — ' 3z ' 



quae, facta evolutione, abciint in seqiientcs : 



2ds =:(2n ■— i)tdz — zdt ; 



2dt z=. 2nsdz -+- zds ; 

 non diversae ab illis, quas Eulerus ex evolutione potestatls (1 -^ x)" 

 secundum conditiones pioblematis sibi propositi , instituta derivavit. 

 Quomodo aiitem hae aequationes cum seriebus sumraandis cohaereant 

 nunc cuique in oculos incurrit. 



5. 5. Que nunc ex liis aequationibus valores quaesitos s et 

 f per integrationem investigemns (quod ab Eulero in prima cjus 

 solutione ope mulliplicatoris idonei factum est), statuamus 5 zzn tv, 

 et nostrae binae aequationes induent has formas : 



2tdv -i- 2vdt =: {211 — i)tdz ~~ zdt 



2dt z=z ztdv -(- zvdt -+- 2nlvdz 



ex quarum utraque deducitur 



3f ( 2n — l)3a — 2^v . _ ' 



7 2tj -H z ' 



3f 2 nvdz -'r- zdv . 



t ■ 2 — zv ' 



undc sublatis denominatoribus, nanclscimur hanc aequationcm : 



dzli2n — 1)(2 — ZV-) — 2nv(2v -\- s)] =r (zz-]- À}du 

 cujus, utpote ad genus Riccatianarum referendae, resolutio vix spe- 

 rare liceret, nisi integralia particularia assignare valercmus. 



y 6. Taies autem valores pàrticularcs facile obtinebimus. 

 si conslderemus duos casus /; rz 00 et Ji z:z. 0. Pro priore casu 

 nostra aequatio dlircrentialJs fict 



dz[2ni2 — zv) — 2nv{2v -{- z)\ z:= 

 quae, facta reductione, abit in 

 1 — zv — VV ZIZ. 0. 



