James Adam, The Republic of Plato

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κατ̓ αὐτὴν -- αὑτῆς: ‘each of them in accordance just with its own peculiar power,’ i.e. in accordance with this, and nothing else. αὐτήν is ipsam in the sense of solam. Cf. 477 D, where it is shewn that δυνάμεις should be classified on this same principle: also 478 A ἐφ᾽ ἑτέρῳ ἄρα ἕτερόν τι δυναμένη ἑκατέρα αὐτῶν πέφυκεν. The reading κατὰ τὴν αὐτήν—see cr. n.—gives precisely the wrong sense. Schneider and others— perhaps rightly—omit αὐτήν (with Vind. F), while Baiter adopts Hermann's ἄλλην —a very improbable correction. It is best, I think, to follow Schmidt, supposing that αὐτήν was accidentally omitted, and afterwards wrongly replaced.

μᾶλλον δὲ κτλ. Socrates had somewhat awkwardly called δόξα a δύναμις, and at the same time spoken of it as possessing a δύναμις. The present sentence introduces a sort of πάρεργον in which the notion δύναμις is more accurately defined. We may infer that δυνάμεις in the sense of (the intellectual) ‘powers’ was unfamiliar at the time when this section was written. It was perhaps— like ποιότης for example—one of Plato's experiments in language. ‘Faculties’ is, I think, too concrete to be a right translation.