$ TO. Quoniatn in huiusmodi calculis omnia ad uni- 

 cam variabilem reduci solent, si pro hoc efficiendo ponamus 

 cy — tdx et dz ~udx^ sumto dx pro constante 3 erit prima 

 aequatio vt sequitur: 



dt (r — pu) -H du Cpt--q) — 0. 

 At aequatio pro superficie erit p -h qt -i^ ru =z o^ unde cum 



hinc fiat p =z — qt ra , prior aequatio hanc induet 



formam : 



dt (r H- qtn -+- ruu) - du (q^rtu -h qtt) = 0, 

 Porro erit 



/i=ax' (tdu — udt)', g^^dx^du-y hz= dx"" dty 

 tum vero ds* ~ dx^ (i -^ tt -h uu)^ et denique 



^ — - 1d^-h_udu __ qdu — r jf — p^u . pn 



ds i + tt-huu qu-^rt r^:^^ pi — q ' 



§ II. At si malimus quartam quandam variabilem, puta 

 angulum Cp introducere, poncndo axzzztacp; aj — li^Cj: j 3% — i;3Cp; 

 aequatio pro superficie erit pt -h qu -h rv ~ 0. Porro pro 

 litteris f^ g^ h^ habebimus 



/ =: d^p" (udv — vdu) 



^z=a$' (vdt — tdv) 



h = d(p" (tdu — udt) 



hinc ergo erit ft-hgu-hhvz^:^. Aequatio pro linea 

 oxevissima erit ; 



fp -\-gq -hhr =zp (udv — vdu) -{- q (vdt — tdv) 

 -h r (tdu — uct) r=z 0, 

 denique fiet ds"" — d(p' (tt -f- uu -h w) , ideoque 



. aaif _- m + ^?u-*. v^v __ g^^ _ rdu __ r^t — p?,v p3^ _ q^t 



os n^uu^ -w qv -TU rt—ipv ^:=r^ • 



NevaActaA(a^.Imp.Scient. Tom.XV. G $ I^ 



