232 
DISQUISITIONES GENEEALES 
modum primum indolem superficiei curvae exprimendi. Retinendo notationes 
art. 4 insuper statuemus : 
dd W 
dx* 
= P\ 
dd W 
Q\ 
d d W 
d z 2 
d d W 
= P'\ 
ddir 
Q\ 
dd IF 
dy. dz 
da;. dz — 
d x. d y 
ita ut fiat 
dP = P'dx -\-R!'dy -f- Q'dz 
d Q = R'dx-\-Qdy-{-P"dz 
dR = Q"dx+P"dy + R'dz 
P 
lam quum habeatur t = — , invenimus per differentiationem 
RRdt = — PdP+PdP = (PQ'-RP')doc-[- [PP"~RR')dy-\- [PR — RQ')dz 
sive, eliminata dz adiumento aequationis Pdx-{- Qd^-j-Rd? = 0 , 
R 3 dt±= (— PPPH- 2PRQ"— PPP')dtf+(PPP"-f QRQ'— PQR— RRR")dy 
Prorsus simili modo obtinemus 
R s du = (PPP"+ QRQ —PQR — R R R') dcc -j- (— RR Q-\- 2 QRP Q QR) dy 
Hinc itaque colligimus 
R 3 T = — RRF'+ìPRQ"— PPR' 
R 3 U = PRP"+ QRQ"—PQR'—RRR" 
R 3 V = —RRQ'-\-2QRP"—QQE' 
Substituendo hos valores in formula art. 7, obtinemus pro mensura curvaturae k 
expressionem syrametricam sequentem : 
[PP+QQ+RRfk 
= PP[Q'R'—P'P”) + Q Q[P'R'— Q"Q)-\-RR [PQ— R"R") 
+ 2 QR(QR"—PP")-\-2PR[P"R"— QQ ) + 2PQ{P"Q'— R R") 
10. 
Formulam adhuc magis complicatam, puta e quindecim elementis con 
flatam , obtinemus, si methodum generalem secundam, indolem superficierum