 Using Taylor series in several variables and exploiting homogeneity, "f"(λ"a"+μ"p", λ"b"+μ"q", λ"c"+μ"r") can be expanded in two ways as
and
Comparing coefficients of λ"p"μ"n"−"p" shows that
In particular, the "p"-th polar of "C" with respect to "Q" is the locus of points "P" so that the ("n"−"p")-th polar of "C" with respect to "P" passes through "Q".If the polar line of "C" with respect to a point "Q" is a line "L", then "Q" is said to be a "pole" of "L".A given line has ("n"−1)2 poles (counting multiplicities etc.)