| ∂ x 1 ∂ u ⋯ ⋯ ∂ x 4 ∂ u x 1 ⋯ ⋯ x 4 ξ 1 ⋯ ⋯ ξ 4 ∂ ξ 1 ∂ v ⋯ ⋯ ∂ ξ 4 ∂ v | = ± F D ′ − E D ″ E G − F 2 {\displaystyle {\begin{vmatrix}{\frac {\partial x_{1}}{\partial u}}&\cdots &\cdots &{\frac {\partial x_{4}}{\partial u}}\\x_{1}&\cdots &\cdots &x_{4}\\\xi _{1}&\cdots &\cdots &\xi _{4}\\{\frac {\partial \xi _{1}}{\partial v}}&\cdots &\cdots &{\frac {\partial \xi _{4}}{\partial v}}\end{vmatrix}}=\pm {\frac {FD'-ED''}{\sqrt {EG-F^{2}}}}}
Analogamente
| ∂ x 1 ∂ v ⋯ ⋯ ∂ x 4 ∂ v x 1 ⋯ ⋯ x 4 ξ 1 ⋯ ⋯ ξ 4 ∂ ξ 1 ∂ v ⋯ ⋯ ∂ ξ 4 ∂ v | = ∓ F D ″ − G D ′ E G − F 2 {\displaystyle {\begin{vmatrix}{\frac {\partial x_{1}}{\partial v}}&\cdots &\cdots &{\frac {\partial x_{4}}{\partial v}}\\x_{1}&\cdots &\cdots &x_{4}\\\xi _{1}&\cdots &\cdots &\xi _{4}\\{\frac {\partial \xi _{1}}{\partial v}}&\cdots &\cdots &{\frac {\partial \xi _{4}}{\partial v}}\end{vmatrix}}=\mp {\frac {FD''-GD'}{\sqrt {EG-F^{2}}}}}
| ∂ x 1 ∂ v ⋯ ⋯ ∂ x 4 ∂ v x 1 ⋯ ⋯ x 4 ξ 1 ⋯ ⋯ ξ 4 ∂ ξ 1 ∂ u ⋯ ⋯ ∂ ξ 4 ∂ u | = ∓ F D ′ − G D E G − F 2 {\displaystyle {\begin{vmatrix}{\frac {\partial x_{1}}{\partial v}}&\cdots &\cdots &{\frac {\partial x_{4}}{\partial v}}\\x_{1}&\cdots &\cdots &x_{4}\\\xi _{1}&\cdots &\cdots &\xi _{4}\\{\frac {\partial \xi _{1}}{\partial u}}&\cdots &\cdots &{\frac {\partial \xi _{4}}{\partial u}}\end{vmatrix}}=\mp {\frac {FD'-GD}{\sqrt {EG-F^{2}}}}}
dove i segni superiori (inferiori) vanno presi insieme. Sviluppando il valore di d s ′ 2 {\displaystyle ds'^{2}} con le formule or ora ottenute abbiamo infine:
d s ′ 2 = e d u 2 + 2 f d u d v + g d v 2 {\displaystyle ds'^{2}=edu^{2}+2fdudv+gdv^{2}}
dove, con le solite notazioni delle superficie,
e = E + E ′ ± 2 F D − E D ′ E G − F 2 {\displaystyle e=E+E'\pm 2{\frac {FD-ED'}{\sqrt {EG-F^{2}}}}}
f = F + F ′ ± G D − E D ″ E G − F 2 {\displaystyle f=F+F'\pm {\frac {GD-ED''}{\sqrt {EG-F^{2}}}}}
g = G + G ′ ∓ 2 F D ″ − G D ′ E G − F 2 . {\displaystyle g=G+G'\mp 2{\frac {FD''-GD'}{\sqrt {EG-F^{2}}}}.}
con le formule or ora ottenute abbiamo infine:
