Hyphenated compounds are not typographically beautiful or altogether congenial to the English language, but in order to emphasize the historic unity the two volumes addressed to Theophilus the expression "Luke-Acts" is perhaps justifiable.
His point here is that Luke and Acts "are not merely two independent writings from the same pen; they are a single continuous work" (8-9), although then Cadbury appears to backtrack with his very next sentence, making a more modest claim: "Acts is neither an appendix nor an afterthought. It is probably an integral part of the author's original plan and purpose." The "probably" here sounds a little odd, but I guess everything is tentative when you're dealing with texts like this.
The problem with Cadbury's thesis is that he doesn't make much of an argument at the outset for why they should go together so closely.
He begins simply enough, by noting that the common authorship of Luke and Acts is the least questioned problem of authorship in the NT, indicated by both their common addresses and their shared use of language throughout (8).
But then his evidence for reading the two volumes as a single work (9) is based on a couple of parallels that don't quite hold. He argues that the phrase ton proton logon in Acts 1:1 should be translated "in volume one"––a plausible enough claim if it fits the form of the two books––but the parallel he cites is Josephus's Against Apion, which explicitly notes at the end of its first book that it is out of space and will continue in a second volume. Parsons and Pervo (Rethinking, 62f) note that this stands in contrast to the ending of Luke, which hasnarrative intergrity and a sense of closure that leave the reader with no more uncertainty than the original ending of Mark.
Cadbury does claim that Josephus is only one example among others, but then what are the others? The only other ancient text he cites here is Philo's That Every Good Person Is Free, which refers to a former volume that argued "that every base person is a slave." Unfortunately, since that supposed former volume is lost, there is no way to compare it literarily with Luke and Acts.
What Cadbury has shown is that a resumptive preface at the start of the second part of a work was one of the "conventions of ancient writing" (9). But he has not shown that that means Luke and Acts are a single work rather than a shorter work and its sequel. Cadbury's conclusion on this matter is weak and (literally) presumptuous:
Occasionally, no doubt, independent works were addressed to the same patron and referred to one another in termss similar to the examples we have given, but without specific knowledge to that effect the presumption in such cases is that the two volumes are really a single work. The preface of Luke confirms rather than opposes this presumption.
So Cadbury admits that a preface such as Acts 1:1-2 could refer to a different work by the same author rather than a first volume of the same work, but he thinks we need to presume that the two volumes make a single work because of the parallel to Josephus. Unfortunately, however, we have seen that Josephus is quite explicit that another volume is coming, in contrast to Luke, so Cadbury's presumption doesn't really hold. (Cadbury addresses the topic of prefaces at length under literary formalia, 194ff; also see p. 344f)
Other scholars have gone looking for better parallels. Parsons and Pervo (63) note another such suggested parallel, in the transition between books 1 and 2 of Philo's Life of Moses. This one was suggested more recently by Charles Talbert, but Parsons and Pervo again show that it doesn't work, because Philo indicates explicitly at the end of the first volume that he will go on and continue his discussion.
Cadbury claims in a footnote that he has aruged how the preface to Luke "specifically contemplates Acts as well as the gospel" (9). He cites three works. The first two are from The Expositor, a journal that ran from 1875–1925. Cadbury's articles, which he doesn't name here, appeared in June, 1921 and December, 1922 of the biannual journal. (Neither BC nor BU has the journal, but I apparently can find the volume at Harvard's Widener Library, CP 44.35, 8th series.) The other source is pp. 491f of a book of essays called The Beginnings of Christianity volume 2, published in 1920 and edited by Frederick John Foakes-Jackson. (BC and BU don't have this one either. Harvard and PTS both list volume 1 in their catalogs, but neither lists volume 2, so I'm not sure where it would be available.)
Ward Gasque (A History of the Criticism of the Acts of the Apostles, 186f), however, discusses these articles. In the first Expositor article, titled "The Purpose Expressed in Luke's Preface," Cadbury translated he asphaleia as "the facts" rather than "the certainty", and he did not take katechethes as implying that Theophilus was a convert; consequently, he interpreted Luke's preface as presenting the facts as an apologetic to counter misinformation Theophilus may have received. In the second (1922) article, "The Knowledge Claimed in Luke's Preface," Cadbury argues that parekolouthekoti implies not research but experience, meaning that Luke had first-hand knowledge of the events he describes. The upshot of these two articles is that the preface to Luke anticipates Acts, in Paul's defense under trial and Luke's presence in the travel narrative.
A second rationale Cadbury offers for seeing Luke-Acts as a unity is the ready explanation for their separation. The first half looked so much like Mark and Matthew, he surmises, that the early church naturally grouped it with them, placing Acts instead alongside other writings (10). This is a true enough claim for what it is, but we should emphasize that it only removes an objection to the unity of Luke and Acts; it's doesn't offer any actual evidence that they were originally one piece. Incidentally, Parsons and Pervo (12) make the converse (inverse?) argument: because Luke and Acts are never found together within the manuscript tradition, we should suspect that they never had been. Unfortunately, as Joel Green points out in his review (CBQ 57, p. 412), no one knows how Luke's first readers encountered the gospel. I don't believe there's actually any evidence for either book before Luke shows up in a group of gospels.