  We consider $N\times N$ tensors for $N= 3,4,5,6$. In the case $N=3$, it is
desired to find the three principal invariants $i_1, i_2, i_3$ of $\bf U$ in
terms of the three principal invariants $I_1, I_2, I_3$ of ${\bf C}={\bf U}^2$.
Equations connecting the $i_\alpha$ and $I_\alpha$ are obtained by taking
determinants of the factorisation \[\lambda^2{\bf I}- {\bf C} = (\lambda{\bf
I}- {\bf U}) (\lambda{\bf I}+ {\bf U})\] and comparing coefficients. On
eliminating $i_2$ we obtain a quartic equation with coefficients depending
solely on the $I_\alpha$ whose largest root is $i_1$. Similarly, we may obtain
a quartic equation whose largest root is $i_2$. For $N=4$ we find that $i_2$ is
once again the largest root of a quartic equation and so all the $i_\alpha$ are
expressed in terms of the $I_\alpha$. Then $\bf U$ and ${\bf U}^{-1}$ are
expressed solely in terms of $\bf C$, as for $N=3$. For $N= 5$ we find, but do
not exhibit, a twentieth degree polynomial of which $i_1$ is the largest root
and which has four spurious zeros. We are unable to express the $i_\alpha$ in
terms of the $I_\alpha$ for $N=5$. Nevertheless, $\bf U$ and ${\bf U}^{-1}$ are
expressed in terms of powers of $\bf C$ with coefficients now depending on the
$i_\alpha$. For $N=6$ we find, but do not exhibit, a 32 degree polynomial which
has largest root $i_1^2$. Sixteen of these roots are relevant but the other 16,
which we exhibit, are spurious. $\bf U$ and ${\bf U}^{-1}$ are expressed in
terms of powers of $\bf C$. The cases $N>6$ are discussed.
  Keywords: Continuum mechanics, polar decomposition, tensor square roots,
principal invariants, cubic equations, quartic equations, equations of degree
16
