Fisher ratio or F_ratio
Well, check out Wikipedia.

The ratio is defined as:

$$
F = \frac{\mathrm{between-group\ variability}}{\mathrm{within-group\ variability}}
$$

where, using your notation, they say that the between-group variability is:

$$
\sum_{i=1}^{M} (u_i - u)^2/(M -1)
$$

and the within-group variability is:

$$
\sum_{i=1}^{M} \sum_{j=1}^{N}( x_{ij} - u_i )^2 / (N - M)
$$