De comparatione arcuum curvarum irreclificahilmm. 



45d 



erit posito brcvitatis ergo /?/^H-/?(y-»-^) <-f-y5«-f-(y— ^)2a= T, si loco istanim formularum 

 surdarum valores ante reperti substituantur ^ 



TdV 



xxdx {/3-^yx-i- dy) -+- yydy {/S-t-yy-i- 8x) = -^ , existente ut ante t = x-*-y et u = xy. 



Vp 



XI. 



Cum nunc sit x*-i-y* = t* — kttu-b-2uUy erit eliminatis variabilibus x ety 



TdV 



sive 



/3{ttdt — tdu — udt) -f- 7 {t^dt—ttdu — 2tudt-\- udu) -¥- du{tdt — du) = ^> 

 dt {iStt — ^u -^ yt^— 2ytu -\- dtu) — du {^t -^ ytt — yu-^ bu) = ^' 



Cum autem sit du = ~i erit hac facta substitutione 



y-8 ^ 



■ ~{-/3^t-^{y-^8)tt-ydt^.-{y-d)Hu) = ^ = :=^. 



7-8 



dV —tdt ^ „ —ttVp 



sicque ent yp==y^ et F=^^^y 



XII. 



Hinc ergo adipiscimur sequentem aequationcm integratam 

 xxdx r yydy 



k 



5) Jy 



= Const. 



{x-i-y)^Vp 



W^A-*-'^Bx-*-Cxx) JViA-t-^iBy-t-Cyy) 



atque in generc concludimus fore 



rdx(^-t-f8x-t-Qxx) rd y{^-t-f8y-t-Q.yy) ^ f8(x-t- y)Vp (§,{x -t-y)^Vp 



J V^A-i-^iBx-t-Cxx) J V(A-t-^By-+-Cyy) ^"^ * y — 8 i(y — 8) * 



siquidem fuerit = a-i-2^{x-^y) -t-y{xx-i-yy) -^2dxy. Erit autem ex relationibus supra 

 Vp ^._ ^ _ __ . 



£(2jB/3 — Ca) 



,„ — M . Vp -,/ '2A^ — Ba 



assignatis ^^ = ^- sive — = —V ^.^-. . . - 



XIII. 



Ponatur jam in gcnere 



a/^dx 



y"dy 



V(A-t-2Bx-*-Cxx) V^A-i-^iByH-Cyy) 



eritque poncndo T= ,S/3 -*- ^ {y-^d)t-i- ydtt-t- {y — ^)^m, 



dF, 



TdV 



x"dx (/?-*- ya; -*- dy) -i-y"dy (/i -h- yy -i- dx) = —r- 

 at ob x-t-y = t et xy = u habebimus x = — — -\ — — et y= — -^o " * ideoquc 



/i-H yx-*- dy 



2p-*-('i-*-8)t-h-(y — S)V(tt — 4u) 



a ^ 2^-t-(y-i-8)t-(y-S)V(tt-4u) 



p-^yy-*-ox = -^ — ~ — - — ^^ -' 



m 



XIV. 



Oifferentiando autem habebimus 



, dtV^tt-Au^-t-tdt-^idu ^ , 



dtV^tt-Au^-tdt-t-^idu 

 2l/(«-4^ ' 



