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We consider $N\times N$ tensors for $N= 3,4,5,6$. In the case $N=3$, it isdesired to find the three principal invariants $i_1, i_2, i_3$ of $\bf U$ interms 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 takingdeterminants of the factorisation \[\lambda^2{\bf I}- {\bf C} = (\lambda{\bfI}- {\bf U}) (\lambda{\bf I}+ {\bf U})\] and comparing coefficients. Oneliminating $i_2$ we obtain a quartic equation with coefficients dependingsolely on the $I_\alpha$ whose largest root is $i_1$. Similarly, we may obtaina quartic equation whose largest root is $i_2$. For $N=4$ we find that $i_2$ isonce again the largest root of a quartic equation and so all the $i_\alpha$ areexpressed in terms of the $I_\alpha$. Then $\bf U$ and ${\bf U}^{-1}$ areexpressed solely in terms of $\bf C$, as for $N=3$. For $N= 5$ we find, but donot exhibit, a twentieth degree polynomial of which $i_1$ is the largest rootand which has four spurious zeros. We are unable to express the $i_\alpha$ interms of the $I_\alpha$ for $N=5$. Nevertheless, $\bf U$ and ${\bf U}^{-1}$ areexpressed 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 whichhas 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 interms 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 degree16

${\bf U} = \mathbf{C}^{1/2}$ and its invariants in terms of $\bf C$ and its invariants