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On an Algebraic Curve, the sum of the number of coincidences at a noncuspidal point 
is the sum of the orders
of the infinitesimal distances from a nearby point 
to the corresponding points when the distance 
is taken as the
principal infinitesimal.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 131, 1959.