note * page 693 Thomson and Tait's Treatise on Natural Philosoph, new edition, vol. i. part ii. §§ 675–678; or Elements of Natural Philosophy, §§ 646–649.

note * page 696 Theoria Philosophise Naturalis redacta ad unicam legem virium in natura existentium, auctore P. Eogerio Josepho Boscovich, Societatis Jesu, nunc ab ipso perpolita, et aucta, ac a plurimis rsecendentium editionum mendis expurgata. Editio Veneta prima ipso auetore præsenta, et comgente. Venetiis, MDOCLXIII. Ex Typographia Remondiniana superiorum permissu, ac privilegio.

note * page 698 “Homogeneous assemblage of points, or of groups of points, or of iodies, or of systems of bodies,” is an expression which needs no definition, because it speaks for itself unambiguously. The geometrical subject of homogeneous assemblages is treated with perfect simplicity and generality by Bravais, in the Journal de 'Éjtcole Polytechnique, cahier xxxiii. pp. 1–128 (Paris, 1850).Google ScholarPubMed

note * page 699 This means such an assemblage as that of the centres of equal globes piled homogeneously, as in the ordinary triangular-based, or square-based, or oblongrectangle- based, pyramids of round shot, or of billiard-balls.

note * page700 This is the assemblage described in the footnote on § 71 below

note † page 701 This also makes A₂OA₂, A₂OA₄, and A₃OA₄ each obtuse. Each of these six obtuse angles is equal to 180°-cos^-1(l/3).

‡ This is the assemblage described in § 69 below, and used in §67, 68, 70.

note * page 701 See §§ 62–71 below.

note † page 701 Thomson and Tait's Natural Philosophy, 2nd ed., vol. i. part 2, § 680; also reprint of Mathematical and Physical Papers, vol. iii. art. xcii. part 1.

note * page 702 Maxwell,Philosophical Magazine, 1860, and Philosophical Transactions, 1867 and 1878 ; Tait, “On the Foundations of the Kinetic Theory of Gases,” Trans. Soy. Soc. Edin., vol. xxxiii., read May 14 and December 6, 1886, and January 7, 1887.

note * page 707 This will be more easily and not less thoroughly understood from illustrations than from a definition in general terms. Of an externally symmetrical man, the two hands are allochirally similar. Either is the pervert of the other; or they are mutual perverts. Two men of exactly equal and similar external figures would be alloehirally similar if one holds out his right hand and the other his left; homochirally similar if each holds out his right hand or each his left. (We ignore at present the monochiral anti-symmetry of one heart on one side ; of interior structure of intestinal canal not in the plane bisecting the exterior symmetric figure, &c, &c). Looking to § (i) below, we see two tetrahedrons, OPQB, OP′Q′R′, which are equal, and allochirally similar, being parallel perverts, either of the other, or parallel mutual perverts. From every point P of a body or group of points, draw a line through anyone point O, and produce to P′, making OP′ = PO. The group of points (P′) is a parallel pervert of the group (P). The groups (P) and (P′) are parallel mutual perverts. Turn (P′) 180° round any line OK. In the position thus reached, it is the image of (P) in a plane mirror through O, perpendicular to OK. In their present positions they are mutual perverts inverted relatively to the line OK. Mutual perverts are allochirally similar.

note * page 712 There is another closest packing of globes or ellipsoids which has the same density as, and might without careful attention be mistaken for, the closest homogeneous packing. For simplicity think only of globes, and take a plane covered with globes touching one another in equilateral triangular order. Look at the accompanying diagram, fig. 6 of § (55) below, and see that there are two ways of placing a second layer on the first to continue the formation of an assemblage. The globes of the second layer may be placed, all of them over the black dots ( · ) or all of them over the white dots ( ˚ ). But having once chosen the position of the second layer there is no more freedom to choose in adding on layer after layer if we are to make a single homogeneous assemblage. Of the two positions which might be chosen for the third layer we must choose the one in which the globes are not over the globes of the first layer. The position of the fourth layer must be the one of which the globes are not over the globes of the second layer, but are over those of the first layer, and so on.

If on the contrary we place the globes of the third layer over the globes of the first, the globes of the fourth layer over those of the second, and so on, we have a peculiar and symmetrical grouping which was first, so far as I know, described by Mr William Barlow (Nature, December 20 and 27, 1883). This grouping is not one homogeneous assemblage. It consists of two homogeneous assemblages, one of them constituted by the first, third, fifth, seventh, &c, layers; the other the second, fourth, sixth, eighth, &c, layers. The consideration of this peculiar mode of grouping may be of great interest in the dynamical investigations to form the subject of my next communication to the R.S.E. (July 15), and, as Barlow has pointed out, may be of great importance in the theory of natural crystalline structure. I must, however, leave it for the present.

note * page 714 In the compound assemblage of two homogeneous assemblages described in the preceding footnote, there are twelve points of contact on each globe, of which nine are placed as those described in the text for the homogeneous single assemblage, and the remaining three are not “at the other ends of the diameters” as described in the text, but are at the opposite points of the small circle on which lie the ends of the diameters referred to.

note * page 718 At ordinary temperatures the angle is 44° 36′·6 (Phillips, Brooke, and Miller's Mineralogy,§§ 407); and at temperature 300° it is almost exactly 45°. Huyghens must have taken it as exactly 45°, as he gave for the ratio of the equatorial to the polar diameter in the statement of his hypothesis.

note † page 718 The shrinkages to pass from the equilateral triangular pyramid to the pyramid with rectangular vertex and to the triangular pyramid for Iceland spar, will be understood in a moment by remarking that the tangents of inclinations of slant sides to base in the three cases are respectively and 1 ; and therefore the distances of vertex from base are as these numbers, the base being unchanged in the simple shrinkage specified in the text.

note * page 720 Perhaps the simplest way of looking at the affair is found by considering that the elliptic section of each ellipsoid in the plane HKK′H′ must remain constant ; and so also must the horizontal and vertical axes of the elliptic section in the plane of the diagram. Hence, while the principal axes of the elliptic section turn in the manner described in §§ 60, 61, the ellipse itself must remain inscribed in a constant rectangle of vertical and horizontal sides in the plane of the diagram, while the third axis of the ellipsoid, which is perpendicular to the plane of the diagram, remains constant.

note * page 724 An interesting structure is suggested by adding another homogeneous assemblage, marked green; giving a green in the centre of each hitherto vacant tetrahedron of reds. It is the same assemblage of triplets as that described in § 24 above. It does not (as long as we have mere jointed struts of constant length between the greens and reds) modify our rigidity-modulus, nor otherwise help us at present, so, having inevitably noticed it, we leave it.