View source for Conjugate directions

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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|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====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Pogorelov,   "Differential geometry" , Noordhoff  (1959)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Blaschke,   K. Leichtweiss,   "Elementare Differentialgeometrie" , '''1''' , Springer  (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.C. Hsiung,   "A first course in differential geometry" , Wiley  (1981)  pp. Chapt. 3, Sect. 4</TD></TR>
</table>