29 



2 1. lîic observasse jnvabit, cum V dx^ -h d/^ -i^-dz' 

 exprimât elemontLini cnrvac , casu X zn x curvam invcn- 

 tain fore Catenariam, si quidem ejus centrum gravitatis in- 

 fimnm occupât locum inter omnes alias curvas isoperime- 



a d X V X 



tricas; at vero si fuerit Xzz: ,' , ita ut r)y ziz " "" — — et 



■^ n a d X ^' X 



y ( I — vim a a x 



Brach ystochronam. 



dz zzz yy^" — ;, manifcstum' est curvam inventam esse 



y ( I — m m a a x) ' 



P r oh I c m a IV. 



Invenire lincam curvam per ternas coordinatas x, y et 2. 

 determinandam, in qiin hacc formula integralls : 



//(xx + yy -4- zz) (ax^ -+- ay^ + 5z^) 

 maxiimun minimumve adipiscatur valorem. 



S o 1 u t i o. 



22. Ponamus brevitatis gratia V xx -\- yy -[- %%z=iv 



et K 1 -\-pp-h(Jf] := s, ita ut nostra formula integralis 

 évadât fVdx, existente Yzivs, ideoque dV z=. sdv -h vds; 

 at vero erit -^^ -- J--^y^J + -d _. ^^ ^^—PdP_-pd3_^ ,,nde se- 



quitur fore M=^; N = -| ; N' :=: ^^ P == V^ P" — '^. 



23. Hinc igitur , cum fiat : 



S — V — P p — P'9 ~ ^Cfi — Pf-^g) _ ^^ 



ternae aequationes nostrae erunt : 



1. 1^-a.^, II. i2L^^ — a.!iç et m. i^i^-d^-^. 



V s ' V s V s 



