O’Sullivan & Hilliard’s The Law of Contract (2018 8 ed). p 65
3.53 As regards the second limb, Smith v Hughes said that actual knowledge is required. The
fact that the claimant should have known the defendant meant to contract for old oats
would not be enough. This is the problem with Smith v Hughes. The principle of objective
interpretation suggests that in order to work out whether the defendant was agreeing to
buy old oats, we look at how things ought to have appeared to the claimant. So Smith v
Hughes is out of line with orthodox rules of offer and acceptance.
In this book on philosophy, the word limb seems to be used in the sense of one of the legs (i.e., supports) of an argument.
Your excerpt is frim page 64 of the book The Law of Contract. On page 63 of that book, we can read “A comment can be made on each of the conditions laid down in Smith v Hughes. With regard to the first condition …”. It seems to me that the author’s discussion of the first condition is the first limb of his argument. If I am right, I think it would been better to have designated it as such. As things are, he introduces a second limb with apparently no reference to a first limb