(144) 



l^a autem proprictas feqiienti qnoqiie ratlone demonftratlir: 

 quia eft /K* — FK*:=/P^ — FP% erit quoque (/K-HFK) 

 (/K-FK) = (/P-+-FP(/P-FP), vnde colligitur CA:CF 

 :r:2CP:/K-FK, et quum fit C A (/K -^ F K) = aC A*, 

 erit omnino C A (/K ^- F K) — C A (/K — F K) — 2 C A' 

 ^aCP.CF,- ideoque CA^FK — CA^ — CP.CF ec 

 CA./Ki::iCA*-i-CP. CF. 



§. 5. Lcmma IV. Si m Ellipfi ducatur corda quaecunque 

 N M cuhis pun6la extrema N, M cum foco F iungantur Uneis 

 re£iis F N , F M , atque haec corda in G fuerit bife&a et per 

 G ducatur G P normalis ad axin maiorem^ iftaque EUipf oc" 

 currat in punCio K et iungatur F K ; erit 2 F K ~ F M -|- F N. 

 Ex pundis N, M in axem CA ducantur perpendiculares 

 NN", MM", et iungatur FK, tum ob NN^ GP et MM^pa- 

 rallelas inter fe N' P' = P M^ quia eft NGmGM, hinc- 

 que colligitur ^FPnrFM'' — F N^- at per Corollarium Lem- 

 matis praecedentis eft FN . C A = CA' — CN' . CF; FK.CA 

 ^CA^— CP.CF et FM.CA=::CA^-hCM^CF; 

 liinc fiet (F M --h F N) C A zrz 2 C A'H- (C M''— C NO C F. 

 At eft C N'' =: C P -4- N'' P et C M' =: P M" — • C P , vndc 

 ob N'' P ~ M' P , erit C N'' ~ C M' =: 2 C P , hincque. 

 (C N-' — C M'') CF=2CP.CF; crit itaque (FM-+-FN) C A 

 — 2 C A' — (C N' — C MO C F rz: 2 C A* — ■ 2 C P . C F 

 ~ 2 F K . C A, et proinde 2FK~FM-f-FN. Tum ve- 

 ro ii ex punifns N, K et M ad Ellipfin ducantur normales 

 N /, K^, M ^a, quae axi maiori in v^ ^, p. occurrent, eadem 

 propofitio hunc quoque in modum demonftratur: ob C N'' — ■ 

 CM'— 2CPet Ck:CN^=z:C?:CP — C|j(.:C M', fiet 

 quoque Cj'— C[x~2Cf, nec non v f ~ f ;jl. Ex quo de- 

 ioceps colligitur Ffx-hFy— ^F^, hinc vero ob Fv:FN 

 :=rFf:FK:zzF|Jt.:FM5 omnino concluditur 2 F K =:: F M 

 -4-FN. §. 6. 



