For those unfamiliar with the facts of the case, see here.
Paragraph 51 of the lead judgment sets out the principles to be applied:
"In summary, therefore, the following are the principles applicable in a case such as this, where a family home is bought in the joint names of a cohabiting couple who are both responsible for any mortgage, but without any express declaration of their beneficial interests.Here, the intentions of the parties did change, and it was logical to infer that Mr Kernott's interest crystallised in 1995 (paragraph 48). The calculation of their shares on this basis produced a result so close to that produced by the judge that it would be wrong for an appellate court to interfere (paragraph 49).
(1) The starting point is that equity follows the law and they are joint tenants both in law and in equity.
(2) That presumption can be displaced by showing (a) that the parties had a different common intention at the time when they acquired the home, or (b) that they later formed the common intention that their respective shares would change.
(3) Their common intention is to be deduced objectively from their conduct: “the relevant intention of each party is the intention which was reasonably understood by the other party to be manifested by that party’s words and conduct notwithstanding that he did not consciously formulate that intention in his own mind or even acted with some different intention which he did not communicate to the other party” (Lord Diplock in Gissing v Gissing  AC 886, 906). Examples of the sort of evidence which might be relevant to drawing such inferences are given in Stack v Dowden, at para 69.
(4) In those cases where it is clear either (a) that the parties did not intend joint tenancy at the outset, or (b) had changed their original intention, but it is not possible to ascertain by direct evidence or by inference what their actual intention was as to the shares in which they would own the property, “the answer is that each is entitled to that share which the court considers fair having regard to the whole course of dealing between them in relation to the property”: Chadwick LJ in Oxley v Hiscock  FAm 211, para 69. In our judgment, “the whole course of dealing … in relation to the property” should be given a broad meaning, enabling a similar range of factors to be taken into account as may be relevant to ascertaining the parties’ actual intentions.
(5) Each case will turn on its own facts. Financial contributions are relevant but there are many other factors which may enable the court to decide what shares were either intended (as in case (3)) or fair (as in case (4))."
The full judgment may be found here, and a press summary, here.