92 Ffaff 



et substituendo \//'f :=F'(x-, y, z, u, iv)^ex j^qaatHone (2.),, fit ,; 



JF'x . dx + Fy . dy + F'z . dzj ^ f/i ^d^j-lj^.^ >4P,.t=>. .;•) '^ ( ^ 



(F (x, y, z. 11, p) . (Fx.d X + f y. dy + f z.-dz ,KJa.dii,+tf p.d?)+ •-/'' p d i) 



vel4us5.95dK.+ ady-f-9{d7,4-®dpr'uh; :- '.\ S' ,"■ 1 



F X — ' t (x, y, z, u, p ) . f X 



y = ri; '' • ■:,■ — = — -rr-H, 



f u . F (x, y, z, u, p) -^ F'n , 



I 



.^ F'y — F (x, y, z, ii, p) . f 'y 



1 -- -11 1 1 * . ■ : ., . ...1 



Fu . F (x, y, z, u»p) — F'u 



r z — .F(x,y,2f,'tt/'py.yy: 



-^ -7- •■ ..■' • . ' 



f ' u , F. ^x, y, z, .11, p) — F' u ^ 



'1 • 



■>'[: 



9i = 



Fp — F (jc; y, i<', u. p) ; Fp '~ '•-//' p • • - --f' 



V5 = ^r— — r — ' , ;.j: , ....;.; *r~r:t j.;. "i -.nii ?i' 



f u ..]f,(Bf,y,7;,i^p>-^E^^rr oi^.wicy 



Functiones liiteris F', f denotatas iüdem pro functionibus datis 



Twf X, y, z, u, p, habendns esse, in aperto est. Jatn vi integrationis Ope prae- 



cedentis problematiS iriveiital&, po§ito p conitante -Vel' dp =='<), aequatio 



du = ^dx -f- Cldy -j- 9^dz consentiie de^bet cum aefpwtione proposita 



■du = Pdx + Qdy + Rdz. Inde aequaiiones 5) := P, D, = Q,9i = R, 



identicae esse et pro quovis valore constantis arbitrariae p valere debent; 



ubi nunc perinde est, sive haec qnantitas indeterpiinata tanquam yariabilis, 



*ive tanquam cohstans consideretnr. Quareut aequatio du = ^dx -j- D.dy 



J^ 9{dz + ©dp cum aequatione proposita du = Pdx+ Qdy + Rdz-j-Sdp 



ex omni parte conspiret, nil aliud requiritur, qxiam ut sit insiiper <5 = S : unde 



1 ' 'i 



prodit F'p — F (x, y, z, u, p) . f p — ^//p ==; S,„£f.u .F. (3?, y,;j,.u,.pj!'-r F'u), 



vel 4^p =^ Fp — F (x, y, «,at, p)'.'fl»''^ S V'Pft'-. F<x,'y, zi'6',l)f 4^S'. Fu. 



Oft LiJ)! "1 jni/r, 11/1 iJ' 



Sic igiiur etiam vp'p aequaljs rep^ritur, functioni datae "riöp x,y,zy.u, p, 



quam littera F notemus. Qnare tandem integratio coinpleta aequationis pro- 

 positae »ystemate harufiijtri,i»jn ,p^q,uatiünura comprehenditux: 



