H. L. A. Hart (1907-1992) practiced law for a time before WWII but after the war taught at Oxford, first in philosophy and then in law as Professor of Jurisprudence (1952-1968). Philosophical work at Oxford after WWII was heavily influenced by the thought of Ludwig Wittgenstein (1889-1951), and Hart’s work on the philosophy of law is no exception, exhibiting a central concern with language and, especially, with the complexity of its ordinary use.
The Concept of Law grew out of Hart’s lectures in an undergraduate course on the subject at Oxford. His first two chapters present first his own understanding of the task of the philosophy of law and then give an account of the motivation for Austin’s views. He goes on to criticize Austin’s views in chs. 3 and 4 (which are directed, respectively, at the ideas of “command” and “sovereign”). Although chs. 1 and 2 are well worth reading, you can get a pretty good sense of his positive account of Austin by looking at just the last paragraph of ch. 2 (p. 25).
Although Hart’s target in ch. 3 is Austin, he also has in mind the views of others, most notably Hans Kelsen (1881-1973). Kelsen was an Austrian who was influential in Europe before the rise of Nazism. He moved to the United States in 1940 and spent the rest of his life here. He was a legal positivist whose views are roughly midway between those of Austin and Hart. Many of Hart’s references to figues like Kelsen do not appear explicitly in the text but instead in his notes at the end, pp. 277-307. Hart will not refer you to these, so you will have to seek them out; and it can be worth doing so since many go well beyond bibliographic references to explain more fully points made in the main text. (If you have the third edition, you will find a further set of notes after Hart’s, pp. 309-325, by the editor of this edition; these concern more recent discussions of Hart and the issues he addresses.)
• Hart’s main point in ch. 3 § 1 is that, if we think of law as consisting of rules that impose duties, we may miss other sorts of law, most importantly rules that confer powers. In subsections of this section, Hart discusses examples of the latter sort of rule, considers two attempts to reduce rules of that sort to duty-imposing rules (i.e., to show that they are only apparently different from duty imposing rules), and considers what can be said against even attempting such a reduction. Look at the range of examples surveyed on pp. 27-33 but also at the explicit comparison of the two sorts of rule in the final paragraph of the subsection (pp. 32f).
• Hart goes on in three subsections (pp. 33-35, 35-38, and 38-42) to describe and respond to attempts by Austin and Kelsen to reduce power-conferring rules to duty-imposing rules.
• The material in Austin that Hart has in mind does not appear in the selections you have read, or even in the first six of Austin’s lectures. In his note on this (see Hart, p. 286), Hart refers you to lecture 23 of Austin, but you can find a fuller discussion in lecture 27 (Austin, pp. 505f). (Austin’s full set of lectures is available on the text browser on Moodle.)
• Kelsen’s ideas on law are summarized in a paper that he published shortly after moving to the United States: “The Pure Theory of Law and Analytical Jurisprudence,” Harvard Law Review (vol. 55, , 1941), pp. 44-70 (on JSTOR at 1334739). He doesn’t explicitly state there the view Hart speaks about, but the ideas it is based can be found in sections IV-VII, pp. 58-63, of this paper.
Although we will probably have very limited time to consider these subsections of §1 in discussion, I encourage you to look at them. In particular, Hart’s response to Kelsen’s idea in the third subsection is very helpful for understanding Hart’s view of the law.