Mistake in Karl Pearson's 1900 paper introducing the chi-squared distribution

Question

Background

I'm reading Karl Pearson's 1900 paper titled:

On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling

I paraphrase the opening. He supposes $x_1, x_2, \ldots, x_n$ is a system of random variables following a multivariate normal distribution centered at the origin. Then he asserts the probability density of the vector $\langle x_1, x_2, \ldots, x_n \rangle$ is $$ Z = Z_0 \exp\left( -\frac{1}{2\det S}\sum_i \sum_j S_{pq} x_p x_q\right)\,, $$ where

• $S$ is the covariance matrix of $x_1, x_2, \ldots, x_n$
• $S_{pq}$ is the "minor obtained by striking out the $p$th row and $q$th column" of the covariance matrix
• $Z_0$ is an unspecified constant.

This appears inconsistent with the multivariate normal density function from Wikipedia, which involves a matrix inverse. These two expressions for the density function can be reconciled using Cramer's rule, but only if what Karl Pearson meant by "minor" is what we today would call a "cofactor."

Question

In 1900, did the word "minor" refer to what we would call a "cofactor"?

Answer 1

Carlo Beenakker was correct. (Thank you.) The formula given by Pearson is wrong (assuming no differences in terminology). The paper should say "cofactor" instead of "minor."

Take $n=2$. Choose an arbitrary positive semidefinite matrix $M$ to use as the covariance matrix of $\langle x_1, x_2, \ldots, x_n \rangle$. Numerically compute the covariance matrix $N$ of the distribution described by Pearson. Observe $N\neq M$.

Code here: https://gist.github.com/Chris-Kimmel/0e9f1e905034dbe2092694cb580787ce