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DISQUISITIONES GENERALES
atque perpendimus, valores differentialium di, du sic prodeuntium, aequales
esse debere, independenter a differentialibus doo, dy, quantitatibus Tdx-\~ Udy,
Udx-\- Vdy resp, inveniemus, post quasdam transformationes satis obvias;
C 3 T = a A d B b'b'+ y Cb'b'
— 2a , Abb , — 2d , Bbb , — 2-f , Cbb'
+ a"A b b + d”B b b + y" C b b
C 3 U — —aAab'—dB ah'—y Cdb'
-f- a!A ((a b'-\~b d) -{- d 'B [a b'-j- bd)-\-y'C (a b'-j- 6 d)
— aAab — d"Bab — y "Cab
C 3 V — a-¿4 5 aV+y CaV •
— 2 a'y4a d— 2 d'Bad— 2y'Cad
-\-a Aaa-\-d"Baa-\~YC aa
Si itaque brevitatis caussa statuimus
Aa -\~Bd A~Cy — B (1)
Aa’+Bd'+Cj = D' (2)
A a"-f- B d "+ Cy" = B” (3)
fit
C 3 T — Db'b'~ 2 l)'bb'A- B"bb
C S U = -Ddb'A- B\ab'A-bd) — D"ab
C 3 V — Ddd—2 B'ad-\~B"aa
Hinc invenimus, evolutione facta,
C\TV- UU) = (BB"—B'B')[ab'— bd)~ = [BB"-B'B')CC
et proin formulam pro mensura curvaturae
j DD"—D'D'
c (AA+BB + CC)*
It.
Formulae modo inventae iam aliam superstruemus, quae inter fertilissima
theoremata in doctrina de superficiebus curvis referenda est. Introducamus se
quentes notationes :