  In our notice we propose the classification of some quartic equations with
only trivial solutions by the auxiliary equations. For proving trivial
solutions of the quartic equations we use method infinite descent based on the
number of prime integers of the solution of the auxiliary equation. The quartic
equations with only trivial solutions belong to "higher arithmetic". The main
tools of their investigation are the fundamental theorem of arithmetic,
infinite descent and elliptic curves. In 1969, L.J. Mordell has observed that
the triviality of the solution of $x^4 + y^4 = 2z^2$ implies the triviality of
the solution of $x^4 + 6x^2y^2 + y^4 = z^2$. In his treatise, he did not study
the reason or the mechanics of this relationship nor did he conjecture that
this link extended to other fourth-degree diophantine equations. We have
discovered that the connection identified by Mordell can be described by using
the concept "resolvents". In present notice which carries an expository
character, we show our findings by presenting the reader with a very easy to
follow and transparent example of how the proposed technique works.
