That "Nature may be conceived to alter her course, without a contradiction," is the material proposition in both Essays; used as an argument to prove, that it is "custom" only which forces the "imagination" to fancy there is a "necessary connexion between Cause and Effect," with a liveliness, and vivacity of conception, equal to a firm belief founded on reason. In the Essays, the whole of these notions are supposed to derive their support from the argument, that as we have no knowledge, either a priori or a posteriori, concerning the "secrets of Nature;" so our observation of the action of a Cause, affords no ground for the conclusions of reason respecting it.
That the idea of causation is only derived from custom, becomes therefore the premises from which the conclusion is deduced, that "beings can begin their existences of themselves;" which proposition, though not formally repeated in the Essays (and which immediately renders void that for the necessity of a great first Cause, and "productive principle" of all things), must tacitly in these Essays be considered as well grounded , because, as every foundation whatever, for supposing any cause necessary for any effect, is denied, and only an influence of "custom on the imagination" is allowed as suggesting a "fancy of it;" it necessarily follows, that nothing beyond what this influence suggests can be assigned as any reason why there should be any productive principle for all the contrivances and ends that take place in the universe; it must therefore, I think, be understood that this "juvenile reasoning" was adopted, and acknowledged but too surely, in the latter Essays.
[Lady Mary Shepherd, An Essay on the Relation of Cause and Effect, pp. 18-20.]
This is perhaps not the most lucid summary. Roughly, Shepherd sees the following progression in Hume's thought:
(1) We have no knowledge, either a priori or a posteriori, concerning the secrets of nature.
(2) Our observation of the action of a cause affords no ground for conclusions of reason concerning it.
(3) Nature may be conceived to alter her course without contradiction.
(4) Therefore the idea of causation is derived only from custom.
(5) Therefore there is no foundation (other than custom of the imagination) for supposing any cause necessary for any effect.
(6) Therefore 'beings can begin their existence of themselves'.
(7) Therefore there is no assignable reason beyond this influence why there should be any productive principle for all the contrivances and ends that take place in the universe.
(8) Therefore there is no need for a first cause and productive principle of all things.
(2) is supposed to follow from (1); (3) from (2); (5) follows from (4); (6), I take it, is supposed to follow from (3) and (5); (7) from (6); and (8) from (7) -- all with suitable additional arguments, of course. Despite the sound of it, Shepherd is not attributing to Hume the position that some beings are self-caused in saying 'beings can begin their existence of themselves'; it is clear from the way she argues against it that she merely means 'beings that can begin to exist without a cause'.
Shepherd will concentrate a lot of attention on the foundation of the argument (1)-(3), arguing that while particulars of causal inference can only be derived from experience, the general form of causal inference can be known by reason a priori. She has a very clever set of arguments for this, which I have briefly touched upon occasionally, which start from a type of cause Hume doesn't discuss much, namely, the cause why something remains as it is or changes the way it does. This is a constituent cause, an intrinsic cause rather than an extrinsic cause; and one of Shepherd's clever moves is to point out both that these causes admits of a type of a priori knowledge by reason that Hume denies of causation and that what extrinsic causes do is add or remove these intrinsic causes. If both of these are accepted, however, Hume is wrong -- at least some causal inference can be justified (as to its general form) by reason alone, however much the details may depend on experience and custom. For instance, on Shepherd's view the difference between mathematics and mathematical physics is not that the former has more certain inferences than the latter, but that the latter has less certain starting-points. The inferences, in fact, are exactly the same. The only difference is that in mathematics you can stipulate your starting-points, and therefore you can always be certain you've found them all. But in mathematical physics your starting-points have to come from without, and often can only be discovered slowly, by measuring, experimenting, testing, etc. And although our ordinary causal reasoning about the world is not as rigorous, either as to inference or as to starting-point, as physics, it is nonetheless underwritten by the same basic logical principles. We can't conceive nature to change her course, at least without supposing a cause for it; because, on Shepherd's analysis, that would involve a contradiction. Shepherd's arguments for this are varied and complex, ranging from positive arguments for her approach to causation to negative arguments against Hume's claims to the contrary. Although this foundation (1)-(3) is perhaps Shepherd's major target, she also explicitly addresses some of the other links in the argument. She has at least two arguments, for instance, criticizing (6) on its own.
Another feature of Shepherd's summary that might need some comment is her reference to 'juvenile reasoning'. The phrase is not an insult, but is a paraphrase of Hume's own Advertisement to the Treatise. After the Enquiry came out, it was still common for people to criticize the Treatise as the more substantial and thorough work. Beattie, for instance, primarily focuses on the Treatise. As an answer to Beattie and others Hume appended a note to future additions of the Enquiry, which has become notorious in Hume scholarship because it is both vague in itself and a complication in the quest to interpret Hume correctly. In it Hume says that most of the principles and reasonings in the Enquiries are found in the Treatise; but the latter was a work whose basic plan had been laid out before Hume left college. Because of this he calls it a "juvenile work," and presents the Enquiries as a recasting of the basic arguments of the Treatise, "where some negligences in his former reasoning and more in the expression are, he hopes, corrected." Shepherd thinks that on the question of causation, the argument in the Treatise and that in the first Enquiry are basically the same. However, she recognizes that she has to justify this assumption; and this is one of the functions served by the digest of extracts she provides. Strictly speaking she probably doesn't need to do so, because most of her argument is clearly directed against the Enquiry, and in any case, as she points out, the Treatise is still being published whether Hume agrees with it or not, and thus its reasoning is still fair game. However, the Treatise has interesting developments on its own that she occasionally brings in (e.g., with regard to beginning of existence), and she's still careful to argue her point that the basic reasoning in the two is the same. (The primary difference, she suggests, is that the Enquiry adds some positions on the application of Hume's causal theory to "the affairs of ordinary life" for the purpose of advocating moderat skepticism.)
