...another's good may be reckoned as being one's own evil, in so far as it conduces to the lessening of one's own good name or excellence. It is in this way that envy grieves for another's good: and consequently men are envious of those goods in which a good name consists, and about which men like to be honored and esteemed, as the Philosopher remarks (Rhet. ii, 10).
The second he derives from ST 2-2.36.2:
...we grieve over a man's good, in so far as his good surpasses ours; this is envy properly speaking, and is always sinful, as also the Philosopher states (Rhet. ii, 10), because to do so is to grieve over what should make us rejoice, viz. over our neighbor's good.
On the basis of these passages, he argues that Aquinas has two definitions of envy, each of which only covers part of the ground: sorrow over another's good insofar as that good diminishes one's good name, and sorrow over another's good insofar as that good surpasses one's own. You could apply one without the other applying. I am utterly unconvinced by his argument. As Perrine himself notes, Aquinas himself does not present these as two definitions. It also requires reading 'good name or excellence' in the first passage strictly as reputational, and 'good that surpasses' in the second passage as not including reputation, neither of which, I think, is plausible. Perrine seems to get his result by taking the first passage to link envy to vainglory and the second to pride, but all of the capital vices, by virtue of being capital vices, are linked to pride in one way or another, and while envy is like vainglory in being a spiritual vice concerned with excellence, and typically derives from it, they represent two distinct areas of moral temptation (vainglory one's own perceived excellence and envy another's). The link is there, but it simply doesn't seem to have the implications Perrine assumes.
In any case, Perrine goes on to argue that to define envy, we need to consider comparative self-worth -- the evaluation of one's own worth relative to another's. When this comparative evaluation concludes that another's worth exceeds one's own, and thus that one's own worth is inferior, then, if a person sorrows at (is pained at, we might say) this, then we have the act typical of the vice of envy.
I think there are two problems with this:
(1) What Perrine is actually describing is jealousy or zeal (they are the same word in Latin), in which you sorrow not over the fact that someone has a good but over the fact that they have it and you do not. Envy is, as Aquinas and the tradition set it up, a kind of sorrowing over another's good in the sense of sorrowing over their having of it, in particular insofar as this can be taken to diminish your own glory or excellence. That is, jealousy is concerned with your lack, envy with their surplus. Perrine denies that zeal involves perception of inferiority, but a perception of inferiority is just a perception of lacking worth in comparison to something else. But envy is specifically concerned with treating the other person's good as a bad thing because it surpasses one's own; this is explicitly stated by Aquinas, and is essential to his account of why it is a capital vice. If anything, it would not be perception of one's own inferiority but perception of another's superiority that would be relevant here, and the two while linked are not the same when it comes to their role in our choices. It's true that zeal doesn't necessarily involve perception of inferiority in self-worth, but sorrowing at another's good because one lacks that superiority is certainly zeal.
(2) It seems false as a matter of moral psychology to say that envy is always based on perception of inferiority of self-worth, because it seems quite clear that one's perceptions of being inferior in self-worth are often caused by envy, not vice versa. There seems no reason to think that envy deals only with self-worth; it deals with any of your neighbor's goods that can be seen as excelling your own in some way, and sees their excelling yours as itself something bad. On the basis of this, one may well count oneself inferior in self-worth. But someone without envy would not necessarily make the move from 'this person has something better' to 'therefore I myself am inferior to them'. The former is about a particular good; the latter is about one's entire self-worth. These are not the same. But in eyes that envy that particular good it can sometimes seem so: their excellence in this is seen as bad for you, period, and focusing as envy does on this particular comparison, it's easy to conclude that their excelling is making you less. This kind of inference is not usually an inference a reasonable person would make; it's an inference that is only plausible to someone whose reasoning is distorted by something like envy. A reasonable person would recognize that their self-worth (the value of oneself) is not less because another excels them in this or that particular valuable feature. That envy does distort our reasoning on such matters, seems quite clear, so Perrine's suggestion seems to put in the formal cause of envy something that is at least often merely the effect.