I was reading Ramsey's 1927 paper "A Contribution to the Theory of Taxation" and came across the following paragraph:
"We have $\lambda_1 = \mu_1,\ldots,\lambda_m = \mu_m$, $m$ hyperplanes ($n-1$ folds) whose intersection is a plane $n-m$ fold which we will call $S$. $S$ will cut the hyper-ellipsoids $u = constant$, $R = constant$ in hyper-ellipsoids which are similar and similarly situated and whose centres are the points $P'$, and $Q'$ in which $S$ is met by the $m$-folds through $P$ and $Q$ conjugate to $S$ in $u = c$ or $R = c$."
Here $u$ is a quadratic function such that $d^2u$ is a negative definite form. $P$ is the centre of the ellipsoid $u = constant$.
Does the term "$n-1$ folds" here just mean an $n-1$-dimensional subspace? And does "conjugate" here just mean "orthogonal"? These are the only interpretations of the terms that make sense to me; I would like to just confirm whether these interpretations are correct...Thank you!