I am trying to understand two papers by James Joseph Sylvester:

P92: *"Note on the properties of the test operators which occur in the calculus of invariants, their derivatives, analogues, and laws of combination; with an incidental application to the development in a Maclaurinian series of any power of the logarithm of an augmented variable."*

and

P95: *"On the multiplication of partial differential operators."*

[The numbering is from volume 2 of Sylvester's Collected Works. These, incidentally, are spread over four volumes: volume 1, volume 2, volume 3, volume 4 (first two courtesy of anonymous book scanners), with some duplicates on the Internet Archive. All of them are out of copyright.]

On the first page of P95 (aka page 11 of the linked two-paper PDF), Sylvester states that

"If $\phi$ be any such function [i. e., a polynomial or power series in infinitely many **commuting** variables $x$, $y$, $z$, ..., $\delta_x$, $\delta_y$, $\delta_z$, ... (here, $\delta_x$, $\delta_y$, $\delta_z$, ... are just symbols, not differential operators!) which is multilinear with respect to $\left(\delta_x,\delta_y,\delta_z,...\right)$], [we have]

$e^{\displaystyle t\phi\star} = \left[e^{\displaystyle \left(e^{\displaystyle t\phi\star}-1\right)\phi}\right]\star$."

Here, as far as I understand, the $\star$ operation is defined as follows (see page 1 of P92, aka page 1 of the linked PDF): If $\psi$ is any polynomial or power series in infinitely many variables $x$, $y$, $z$, ..., $\delta_x$, $\delta_y$, $\delta_z$, ..., then $\psi\star$ means the differential operator we obtain if we collect all the $\delta_x$, $\delta_y$, $\delta_z$, ... variables at the right end of every monomial and replace them by the partial derivative operators $\frac{\delta}{\delta x}$, $\frac{\delta}{\delta y}$, $\frac{\delta}{\delta z}$, .... I cannot say that I am sure about this, though, because no matter how I try to obtain a small, verifiable example for the formula, I get some nonsense which is either wrong or I am not able to check.

Sylvester studied these in the context of classical invariant theory, but nowadays quantum field theorists are interested in these differential operators as elements of the Heisenberg algebra. Is there any modern (readable) reformulation of the above identity? Has anyone else tried to comprehend its meaning? Is it related to the identity $\left(\exp a\right)\left(\exp b\right)\left(\exp a\right)^{-1} = \exp\left(\left(\exp\left(\mathrm{ad} a\right)\right)\left(b\right)\right)$ which holds for any two elements $a$ and $b$ of a ring for which these exponentials make sense? (This is speculation based on nothing more than the appearance of nested exponentials in both identities.)