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A formula relating the differential of the [[Normal|normal]]  $  \mathbf n $
to a surface to the differential of the [[Radius vector|radius vector]]  $  \mathbf r $
of the surface in the [[Principal direction|principal direction]]:

$$ 
d \mathbf n  =  - k _ {1}  d \mathbf r \ \ 
\textrm{ or } \ \ 
d \mathbf n  =  - k _ {2}  d \mathbf r ,
$$

where  $  k _ {1} $
and  $  k _ {2} $
are the principal curvatures.

The formula was obtained by O. Rodrigues (1815).

''A.B. Ivanov''

A representation of [[Orthogonal polynomials|orthogonal polynomials]] in terms of a weight function using differentiation. If a weight function  $  h ( x) $
satisfies a [[Pearson differential equation]]

$$ 

\frac{h  ^  \prime  ( x) }{h ( x) }

 = \ 

\frac{p _ {0} + p _ {1} x }{q _ {0} + q _ {1} x + q _ {2} x  ^ {2} }

 \equiv \ 

\frac{A ( x) }{B ( x) }
 ,\ \ 
x \in ( a , b ) ,
$$

and if, moreover, at the end points of the orthogonality interval the following conditions hold:

$$ 
\lim\limits _ {x \downarrow a }  h ( x) B ( x)  = \ 
\lim\limits _ {x \uparrow b }  h ( x) B ( x)  =  0 ,
$$

then the orthogonal polynomial  $  P _ {n} ( x) $
can be represented by a Rodrigues formula:

$$ 
P _ {n} ( x)  = \ 
c _ {n} 
\frac{[ h ( x) B  ^ {n} ( x) ]  ^ {(n)} }{h ( x) }
 ,
$$

where  $  c _ {n} $
is a constant. Rodrigues' formula holds only for orthogonal polynomials and for polynomials obtained from the latter by linear transformations of the argument. Originally, this formula was established by O. Rodrigues [[#References|[1]]] for the [[Legendre polynomials|Legendre polynomials]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O. Rodrigues,   "Mémoire sur l'attraction des spheroides"  ''Correspondence sur l'Ecole Polytechnique'' , '''3'''  (1816)  pp. 361–385</TD></TR></table>

''P.K. Suetin''

====Comments====
For part 1) see also [[#References|[a1]]], [[#References|[a2]]]. For part 2) see also [[#References|[a3]]], [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Darboux,   "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , '''1–4''' , Chelsea, reprint  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.P. Do Carmo,   "Differential geometry of curves and surfaces" , Prentice-Hall  (1976)  pp. 145</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Szegö,   "Orthogonal polynomials" , Amer. Math. Soc.  (1975)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  T.S. Chihara,   "An introduction to orthogonal polynomials" , Gordon &amp; Breach  (1978)</TD></TR></table>