What does $d$ mean in this notation of the "usual noninformative prior of $\mu_i$ and $\sigma_i$?" Samiuddin, (1976) states:

or, typset with $\LaTeX$ as originally posted

We start with the usual noninformative
  prior distribution of $\mu_i$ and
  $\sigma_i (i = 1,2,\ldots, k)$
$$\pi(\mu_1, \mu_2, \ldots, \mu_k;
 \sigma_1, \sigma_2, \ldots, \sigma_k) 
 \propto d\mu_1d\mu_2\ldots d\mu_k
 \frac{d\sigma_1d\sigma_2\ldots d\sigma_k}
 {\sigma_1\sigma_2\ldots \sigma_k}$$

What does this notation mean?

Samiuddin, M. 1976. Bayesian Test of Homogeneity of Variance. Journal of the American Statistical Assoc. Vol. 71, No. 354  
 A: This is shorthand notation for a "differential" of the mean and variance parameters.  The longhand version goes:
$$p(\mu\in[\mu_1,\mu_1+d\mu_1)|I)\propto d\mu_1$$
This indicates a uniform probability with respect to $\mu$.  A more familiar notation is:
$$p(\mu|I)\propto 1$$
It comes from the "proper" derivation of a PDF from a CDF.
$$lim_{dy\rightarrow 0}P(Y\in[y,y+dy))=f(y)dy$$
EDIT: I initially wrote this answer in a hasty fashion, and so had a bit of unclear notation myself.  In my example, I only had a 1-dimension variable $\mu_1$, and all the above relate to a 1-dimensional random variable.  I think the statistical physics literature ("maxent" people) uses this notation (but not entirely sure) - Edwin Jaynes, Larry Bretthorst, Stephen Gull, and others.  I've never seen it explained in any more detail than what I have given.
And second is that $I$ stands for "prior information", not an identity matrix.  This is just a good habit to express $I$ explicitly as part of your assumptions - so that you don't forget that 1) they are there, and 2) you answer depends on the prior information just as much it depends on the data.
