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