These notes are to supplement my paper (4), and should be read in conjunction with it. Both are divided into three parts, and in these notes the section numbers have a further digit added; thus §1.41 here supplements §1.4 of (4). References by section numbers are always to (4) or to the present notes, but references to other papers are numbered independently.
The principal results of these notes are the following. New sphere packings are given in [2
], m ⩾ 6, and in , which are twice as dense as those of §§1.6, 2.3. Others are given in [2
], m ⩾ 5, with the same density as those of §1.6, but in which each sphere touches fewer other spheres than in the earlier packings.