AD CURVAS INVENIENDA S AESOLVTA. 5 ZExemplum VII.

z8- Invenire curvam , in qua fit s(xx -}- yy j n dx y/ (i+pp)maximum vel minimum.

Cum hic, sit Z = (** 4 .^)«'/(i +■/'/')j erit ^Z =

s ^+^)”~ I (y^4-jy^)vci+/>/>) + ^ V+'p py’

ergo 2V = r * (xx-b# (i &P= I

ex quo pro curva qutesita ista habebitur «quatio

zn(xx + yyf~~ 1 ydxs/Ci -{-PP) ~d ( x JL+ i y ± J ._ —

_ V ( i -b??)

Zn(xx-\-yy) n l p (x d x + y dy) . d p ( x x -4- <y <y fi

VCr+Ff) - + o+J$> - *i u * P cr

(xx~\~yy) 1 divilä, ac per \f (i-\-pp) multiplicata, abit irr

X n y d xdp

r+f?

: j „Xd, -j- feu a “W*—»

y r ”b p xx +yy

Hujus aequationis utrumque membrum integra-

bile est per quadraturam circuli, fitque integrale xn A tang-

x

= A tang. p-i- A tang. k — A rang. : unde siet — —tan g — A tang. T ; eritque T" functio algebraica ip-

sius />, dummodo lit 2 » numerus rationalis. Cum ergo sit at7>, seu /= erit dy=pdx= ~— , sivex/F

Tdp

dx xdTT fr

= Ta?x— pTTdxi ideoque — =——^— +r 1 x r - fTT

T - Tdp

unde prodic /*• — /-

- -p T ' 1 l—~pT __ - f.

dem ad construendam curvam abunde iätisfaciunr. Verum ut ha-rum curvarum, qua? pro defimVs exponentis«v aloribus prodeunt,,natura melius cognoicatur , Casus nonnullos contemplabimur.

ll Sit » =i, & 2 n=^ 1 ; erit Atang. — = A tang.

G 3 ' St