IJO ^ [DE USU METHODIS C H 0 L I 0 N II.

22. Eadem hac Exempla omnia quoque resolvi possunt perMethodum supra jam traditam j quare cum utraque via eademsolutio obtineatur, juvabit solutionem per alteram viam unoExemplo exhiberi. Sumamus igitur tertium Exemplum, in quomaximi minimive formula fy x dx xfx d x \/ ( i +pp) > disse-rentiando iterumque integrando per partes, reducitur ad hancformam fy x dx fx dx \' pp) 4- fx dx (14-//0 ßxdx i

cujus utrumque membrum in Casu secundo supra §. 7. expolitocontinetur. Quaratur itaque utri usque valor disserentialis, eo-rum enim summa, posita =0, dabit aquationem pro curvaquasita. Formula autem fyxdxsxdx*J{x-\-pp) cum Casusecundo collata , dabit n =-fx dx\f ( 1 +■/»/>) & Z~-=yxn ;unde fit L -=y x ; M—y n , N=xn, P = o , & c. De-inde erit [Zj = x y/( 1 ~Ppp) i indeque [ M~\ .== \/ (1 -f- pp).

lN] = o, &[F] =cujus valor . posito x

X P

Porro o&fLdx = ßxdx ,

v c 1 4 -ppy

— a, quem generaliter posuimus H,hic in solutione Exempli est A; ita ut sit V = A — fyxdx.Quare hujus formula valor disserentialis erit = n v. dx ( x n

_ _l_ 1 xp(A - / y xdx) ) '

dx V ( t + pp) ^

,, - f ; + i- d. V"*?* ). Altera formn-

V(I +P?) dx V(! 4 -pp)

la fx d x sj (1 -| -pp ) fy x dx, cum Casu secundo §. 7. colla-ta , dat n = fyx dx & Z = x n v' ( i-pp p) , unde erit L= x y/ (1 -h pp)> M=n ( 1 4 -pp), N = o, & P —

/ d.

= nv\ d x (*/* xdx\I ( I -f- pp)xpfyxdx ,

JJf+fsy ^^ ncc l ue f^dx z=ßxd x s/ ( 1 4 -pp): quare cum

H sit valor ipsius fL dx, posito x = a, erit H -= B, & V% = B — fx dx \/( 1 -\-pp ). Porro est [ Z ] — y x, hineque [A/j= 7 5 & £Fj] = o. Ex his prodit valor disse-

rentialis = nv. d x (Bx — xfx dx ^ (i 4 - p p) — j~ X

d.