30 Johannes (I) Bernoulli. 



tiale -dt (NB. suniitur -dt, quia crescentibus z ipsae t decrescunt, 

 et proinde différent, ipsius t est quantitas negativa) 



-2ab z 3 dz 3 +4aab zz dz 3 -b 3 zz dz 3 -2a 3 bz dz 3 + 2ab 3 z dz 3 



zz -2az + aa -bbVzz -2az + aa -bb in 0:z-a 

 Sit itaque aequatio generalis dy 3 = dz 2 dt convertitur in haue 



b 3 z 3 dz 3 

 zz -2az + aa -bbVzz -2az + aa -bb in C : z -a 

 2abz 3 dz -4aab zz dz + b 3 zz dz + 2a 3 bz dz -2ab 3 z dz 

 zz -2az + aa -bbVzz - 2az + aa -bb in D : z - a 

 multiplicata aequatione per zz -2az + aa -bbVzz -2az + aa -bb 

 in C:z-a, et divisa per bzdz 3 , prodibit redueta ad cyphram 

 haec: 2az 3 - 6aa zz + 6a 3 z - Sa bbz - 2a 4 + 2aa bb = 0. Si a = b 

 oritur 2zz -6az + Saa = et z=\ a+ Vf aa=FB. 



[37] Esto nunc ABC (Fig. 30) Conchois altera, in qua nempe 

 singula reetangula FNB eidem FM A su nt aequa lia, et positis 

 quae prius, sit FN = x erit no=dx NM= Vxx -bb 



bdx , T7 _. ab _,„ ab + xx 



No = — -== =— , NB= -, FBseaz= 



Vxx - bb x x 



, ., xx dx -ab dx 



et be seu dz = , 



xx 



jam quia FN • FB :: No • Be erit Be seu 



._ abb dx + bxx dx 



d y= / — 



xxV xx - bb 

 et quia be • eB : : BF • t erit 



aab 3 + 2abb xx + bx* 



t 



x 3 - abxVxx -bb 

 sumptisque diff erentialibus erit 



-dt = -Qabbx 6 -b 3 x 6 + 5ab i x i -4aab 3 x* + 5aab 5 xx + 2a 3 b*xx-a 3 b 6 



dx 



m 



xx -bbVxx -bb in D : x 3 -abx 



Si itaque in aequatione generali dy 3 = dz 2 dt substituantur valores 

 inventi proveniet haec aequatio 



dx 3 in C : abb + bxx 

 x 8 -bbx 6 Vxx -b b 



