458 .mmVA.C. EULERI OPERA POSTHUMAiO^ a(V Analym. 



5. Si in formula priori ponatur a? = c, eril 2/7. c=77. 2cy(i — cc). Ac si porro ponatur 



aj = 2cl/(l — cc), ut sit /7.a; = 2/7.c, erit ob Yii — a;a:) = 1 — 2cc, .ninim (v"i> ji 

 -<;: ;^ .;; - : : ^. - ;r 1 ■ ; i. 377. c = /7 . (3c — ^^c^). ilayioliib nohiRj ui 



i*osito antem Tiltra (i;=ii:3c~ ^ID*; '^^^ . - .. > >.:.-.,;. iaYJenB ni rarn oiJui.^ 



*77. c = /7. (.a^y(i — cc)-i-cy(l — obx)) 

 unde multiplicatio arcuum circularium est manifesta. 



De comparatione arcuum Parabolae. 



* ., .6. Existente (Fig". 55.) AB parabolae axe, sumentur abscissae AP in tangente verticis A, sitque 

 parameter parabolae =2; unde vocata abscissa quacunque AP = z, erit applicata P.p = —j ideoque 

 arcus Jjp =ydzl/(l-i-zz), quae expressio ut ad nostras formulas reducatur, in hanc abit f ^jr--—\' 



Quare fleri oportet A=\, B = et C=l, unde ut ante habebimus 



.i;; !j'ffoo«i iiuiil \'it"'fiu f:'<-jo';;,' . :— V, <' .' . . 'vIhkU; W-mi or^ .toilAnifi/ 



^jt er^o y= 1 et p = cCy atque aequatip relationem inter a? et j exhibens erit j^s^vbr. oBitTjJ 



= — cc-*-aja7-4-jj — 2xyy{cc-i-i), seu y = £C"|/(1 -i- cc) -f- cl/(l -i-cca?). ?'^ur,u 



7. Deinde ob V/} = c et 7 — ^= 1 -h ^(1 -4-cc), facto 2(=1, J9 = et,6— Jl, erit ex 

 formula XII data 



' ' / vto (l -4- aia;) ___ rdy(i-t-yy) f onsL — <^(a?'-*^"y)^ " ' ^ 



./.l/(l-+-4^) J Vii-t-yy) .,.7. a-t^ayaH-cft)* 



At est ajH-j- = a? (1 -i-l/(l -i-cc))-ji-. p '1/(1 -4- aja?), ergo ._ {» 



{x-i-y)^=2xx (1 -Hcc -1-1/(1 -*-cc))H-cc-f-2ca?(l -f- 1/(1 -i- cc)) y(i-i-a;a?): 



Quare formularum istarum integralium differentia erit 



Const. — cxxy{i -\-cc) — ccaiy^"^ xx) = Const. — cx'y:'V^ \^hin\i> .£ 



8. Indicetur arcus parabolae abscissae cuicunque z respondens fdzy{i-¥-zz) per 77. z, et 

 nostra aequatio hanc induet formam: 



77 . a? — 77. {xy{i-¥-cc) -h c1/(1 -\-xx)) = -~n.c — cx (xy{i -h-cc) -4- cl/(l -t-xx)), 

 sive n .c-t- TI .x = n . (a2l/(l-i-cc) -^ dy{i'^-()iix)) — cx{xyii-t-cc) -h- cy{i -t-xx)). 

 J)atis ergo duobus arcubus quibuscunque, tertius arcus assignari potest, qui a summa illorum deficiat 

 quantitate geometrice assignabili. Vel quo indoles hujus aequationis clarius perspiciatur , erit 



n .c -i- n .x = n .y — cxy 

 siquidem tuerit 3^ = a7y(l -f- cc) -+- cy(l -f-aja;). . 



9. Cum sit j > a?, sint in figura abscissae AE = c, AF==x et AG = y, erit arcus ^e=77.c 

 et arcus fg = n .y — 77 . a?; hinc ergo habebimus 



Arc , Ae = Arc . fg — cxy, seu Arc. fg — Arc . Ae = cxy 

 existente y = ajl/(l -1- cc)-hc1/(1 H-a^a?). Ex his igitur sequentia problemata circa parabolam 

 resolvi poteruut. Aiii\m\'A) ,&}ifiup9« ujwujui oiuiouh •»f,iifi'V!Ml*jh ,:*}• 



■ 8*. 



