de noua Tneihudo integrandi. 89 



quijiciue quauiitates X, Y, Z, 9^. SX. deierminare oportet. Calculis rite sub- 



rliuMi.s, prodit Y= •— , exLStenle 



^(PR''_RP>' + RT'— TIV + TP'«— PT") (SP" — PS^'+P"— S') 

 gn =^j — (PR"— RP"-}-RS' — SR'-f Sr — PS'") (TP"— PT"+P'_tO; 

 ( — (PS" — SP'-J-Sr — TS'-f TP" — PT") (RP"— PR"-f F"— R')* 

 t (PR"— RP"H-RS'— SR'-t SP"— PS ') (PQ'— Qp» + QT— TQ'-f-TP-- PT) 



SR = j— (PR'— RP'+RT-TR+TP'"— PT")(PQ"-<?I"''+<?S'-SQ + SP"-PS'') 

 (-J-(PS^— SP^-+.ST'— TS' + TP"— PT-)(PQ"-Or+<2R-RQ'tRP«_PR") 



Ex Y prodit Z, permutando invioem litteras Q et R, atque indices " et *"• 



simili modo ex Y prodit ^ vel d , permutando in expressione pro Y litteras 



Q et S, vel Q et T, nee iion indices " et ", vel " et ". Sic denominator S 2 



invariatiis nianet, nisi quod signa mutet. Quod ad X attinet, inventis 



1 — QY — RZ — Sg> — TQ. 

 Y, Z, ^, Ö, est X = , vel etiam X prodit 



ex Y, permuundo invicem litteras P et Q, indices ' et ". 



Qua ratione detenninatis X, Y, Z, 9>, £>., «i fingendo a, b, c, e, f esse con- 

 »tante«, ex aequationibn» dx = Xdu 



dy = Ydu 



dx = Zdu. 



dp = ^dn 



aq = Q.du 

 secunduin §. a. exprimantior x, y, z, p, q per u et qiiinque konstantes arbitra« 

 rias a, b, c, e, f ex ipsa iniegradone ingressas, eaedem expressiones, suppo- 

 sitis deincfcps a,b,c, e, f, variabilibus, praebebiint functiones ^läv u, a, b c e f 

 ila coniparatas, ut eas pro x, y, z, p, q substituendo aeqnaüo diiFerentiglis 

 proposita abeat in aequationem inter ^juinque quantitates a, b, c, e, f eanun- 

 qiie difFerentialia. Jam vero ex problemate (3) integratio hujos aequationis 

 transformatae his comprehenditur tribus aequatipnibus : 



1) F (a. b, c, e, f ) = ^^ [f (a, b, c, e, f), f] 



2) F (a, b, «, e, f ) = ^' [f(a, b, c, e, f;] 



3) F (a, b, c. e, f)= 4/'f. 



