  In this paper we first show that, under certain conditions, the solution of a
single quadratic diophantine equation in four variables
$Q(x_1,\,x_2,\,x_3,\,x_4)=0$ can be expressed in terms of bilinear forms in
four parameters. We use this result to establish a necessary, though not
sufficient, condition for the solvability of the simultaneous quadratic
diophantine equations $Q_j(x_1,\,x_2,\,x_3,\,x_4)=0,\;j=1,\,2,$ and give a
method of obtaining their complete solution. In general, when these two
equations have a rational solution, they represent an elliptic curve but we
show that there are several cases in which their complete solution may be
expressed by a finite number of parametric solutions and/ or a finite number of
primitive integer solutions. Finally we relate the solutions of the quartic
equation $y^2=t^4+a_1t^3+a_2t^2+a_3t+a_4$ to the solutions of a pair of
quadratic diophantine equations, and thereby obtain new formulae for deriving
rational solutions of the aforementioned quartic equation starting from one or
two known solutions.
