  In this paper we obtain four new parametric ideal solutions of the
Tarry-Escott problem of degree 7, that is, of the simultaneous diophantine
equations, $\sum_{i=1}^8x_i^r=\sum_{i=1}^8y_i^r,\;r=1,\,2,\,\dots,\,7$. While
all the known parametric solutions of the problem, with one exception, are
given by polynomials of degrees $ \geq 5$, the solutions obtained in this paper
are given by quartic polynomials, and are thus simpler than almost all of the
known solutions.
