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feit i ud N (x, y, c) = 0 = n, cum consiante arbitrio c. Irrationalitate sublata formam functiouis integrae induere poterit, postmodumque ejus differentiale οη + 3η.θ^ = 0 rationaliter continebit χ, γ et c, ideoque elix y χ ininato c, tupererit aequatio rational^ inter χ, γ et y (= y), quae seX cundutn γ' altioris baud erit gradus, ac ipsa n — 0 secundum c. Ipsa enim illa aequatio differentiata n' + n^y' = 0 suppeditat y' ralionaliter in c9 ideoque ipsi y' plures valores diversos (pro iisdem a? et y) tribui nequit, ac quae ipsi c ex aequatiooe w = 0 eompetit; quare necessario ipsum γ per aequationem ejus gradus, eujus est n seeundum c, determinabitur; (vel etiam minoris, si c valoribus aequalibus gaudet, quos tarnen facile per θ n = 0 exsulare facias.) Sin vero aequatio n = 0 ipsum c rationaliter uon contineret, ipsa vel secundum c rationalis efficienda esset, vel etiam resolvenda. Si igitur ita obtineretur CSSSSK, seu c = z(ary), exsistente z funclione irrationali vel etiam transcendente quadam ipsorum χ et y, obtineretur statim dfc= 0 = z*dx-^-Zidy, seu aequatio diflerentialis dy-\-pdx = Q, (exsistente z'z^p.z^) quae universim illius functionis surdae vestigia aliqua retinebit, vel etiam aliquauto simplicior evadet. Differentiando enim functionem irrationalem gradus cujuscunque^ obtiuetur omnimodo similis. Si enim esty ejusmodi ipsius x, nempe data per aequationem rationalem tt°(#,y) = 0, erit n'dx-^x^y^^Q^

journal für die reine und angewandte mathematik

Disquisitio, qualis aequatio differentialis gaudeat integrali algebraico completo? qualisve primarie transcendenti? quaenamque forma integrali competat?