Some journals restart their volume numbering on occasion. This reset marks a start of a new "series". In order for a reference to point to a unique paper, it is then common to specify both series and volume numbers (though series can typically also be inferred from the publication year). In the case of Annals of Mathematics, a new series was started the first time the journal moved universities.
The January 1904 Proceedings was the first in the "Second Series". In it on page 1 was a
Notice Concerning the Second Series of "Proceedings."
THE following statement made by the President, Prof. H. Lamb, at the meeting of the Society, held on June 11th, 1903, is reported in Proceedings, Vol. XXXV., pp. 460, 461:—
"The first four volumes of Proceedings contain the papers of eight and a half years (January, 1865–June, 1873). Vols. V.–XXIX. correspond with the Sessions 1873–1898, one with each Session. Vols. XXX.–XXXV. are edited in accordance with the rule that the 'Volumes shall contain as nearly four hundred pages as may be found convenient, provided that each volume shall begin with the Report of Proceedings at a meeting, not necessarily an Annual General Meeting.' These volumes contain the records of proceedings at meetings, followed by the papers read at the meetings, and they also contain Appendices in which are Notes and Corrections and Obituary Notices of deceased members. The Council has decided that in future the records of proceedings at meetings and matter of the kind previously placed in Appendices shall be collected at the beginnings of volumes and shall have a different pagination from that of the papers, and that the records of proceedings at meetings shall be issued for a Session at a time in the earliest completed volume after the end of the Session. … The volumes are to contain as nearly five hundred pages as may be found convenient."
To summarize — the London Mathematical Society changed their rules governing the Proceedings, enough that they decided to clearly separate the Second Series from the original.