  In this paper we generalize the idea of "essentially unique" representations
by ternary quadratic forms. We employ the Siegel formula, along with the
complete classification of imaginary quadratic fields of class number less than
or equal to 8, to deduce the set of integers which are represented in
essentially one way by a given form which is alone in its genus. We consider a
variety of forms which illustrate how this method applies to any of the 794
ternary quadratic forms which are alone in their genus. As a consequence, we
resolve some conjectures of Kaplansky regarding unique representation by the
forms $x^2 +y^2 +3z^2$, $x^2 +3y^2 +3z^2$, and $x^2 +2y^2 +3z^2$.
