397 



jam N x . F .r =./ œ, A . N œ . N^j^a .Çl N(+lè.ptf\' • Est 

 vero fx numerator atqiie Nx deiioniiiiator ipsius Fx\ qui iffitur pro- 

 prietate proposita gaudent, quae, facto [o = yct, atqiie A ^ ct . N o\ \\aGC 



erit N'o\N{x + iß. N{x — y)= [Nxf .{NyY -\- y/x'. /y\ Ex casuy .= n 



vero o1)tinetiir yfx =No^.N2x — Nx , ideoqne /",»; <^ Fx per solam 

 Nx exprimuntur. Si i^itiir luinc valorem introdiixerimus, erit 



[No\ jV^H^ . JVT^y — Nx . Ny']^ (No' N-2X—Nx\{No'N-2y— Ny) 



seu No- . Nx+ij . Nx — y — 2 ÎSfx . Ny . Nx-^y. N'x—y -. No\ N>x.N>y 



— No (N-iX . Ny-{-N2y . Nx), quae aequatio indolem denominato- 



ris N g;eneratim exprimit. (Scribimus vero jam N z pro {N z)" ). 



IV. Numqiiae fmwtio ejusdem imhlis ac lamma datur, numve similis, cujus 

 aequatio fund amen tali s cnmplemento logarit/tmico caret? 

 Sit hujus signum y, idque ejus vis , ut s\ty xy = y x -\-yy — y x — y y'-\-a, 



si x'= X . Y^-^ ^ y'= y r^~. ideoque I — x' = [i — x) : {i —X7j) ^ 



\—y'^{\—y);[\—Xy). 



1) Ut igitur ipsa inveniatur, babeamus primo x' pro Const., ideoque 

 1 — ydx = xi—xdy, cui aequationi satisfit per dx^xi — x <^ 

 dy—\ — y. Quibus positis, fit difTerentiando d xy — xdy -\-ydx = 



yi'l — x) x^x [). — \i) = x[\ — xy), dx'—o, dy' — ^^_^^^^=\—x.\—y\ 



7, ....ry . 1 — X y.x=y, X . X 1 — x + y,y . 1 — y — 7, y'. 1 — ?/. 1 — x-\-oa 



quod ponas =x Z = X-\-Y — 1 — x .fy'-Y ^ a. 



2) Deinde vero habeamus y' — Const., ideoque 1 — xy dy' — o— 1 — x dy+ 

 y (y — i)dx, quare jam dy = y{i — y)^dx=i — x ponere licet, 



fitque ut antea dxy=y. 1 — xy atque dx' = (1 — y) • {^ — '«'), (per- 

 nnitando sc.,r&y). Repraesentata vero mox obtenta aequatione 



breviter per x . Z = X-{- Y — 1 — x • fy' -\r ^ a, ut A' = x- — x"^- y, x, 



Y= 1 — y-y,y sit, atque difïerentiata, fit ^ . 1 — x -^ x xy Z . y 1 — xy = 



51. a 



