Conjugate directions

A pair of directions emanating from a point $ P $ on a surface $ S $ such that the straight lines containing them are conjugate diameters of the Dupin indicatrix of $ S $ at $ P $. In order that the directions $ ( du : dv) $, $ ( \delta u : \delta v) $ at a point $ P $ on $ S $ be conjugate, it is necessary and sufficient that the following condition holds

$$ L du \delta u + M ( du \delta v + dv \delta u) + N dv \delta v = 0, $$

where $ L $, $ M $ and $ N $ are the coefficients of the second fundamental form of $ S $ evaluated at $ P $. Example: a principal direction.

References

[1]	A.V. Pogorelov, "Differential geometry" , Noordhoff (1959) (Translated from Russian)
[a1]	W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , 1 , Springer (1973)
[a2]	C.C. Hsiung, "A first course in differential geometry" , Wiley (1981) pp. Chapt. 3, Sect. 4