We liken the internal heat of the earth to the rudimentary wing-bones of the apteryx. In each case we find a survival devoid of much significance, unless in regard to its historical interpretation. But that historical significance can hardly be over-estimated. Unimportant as the wing-bones may be, they admit of explanation only on the supposition that the apteryx was descended from a winged ancestor. Unimportant as the internal heat, still lingering in our globe, may seem, it admits of explanation only on the supposition that the earth has had the origin which the nebular theory suggests. That the earth’s beginning has been substantially in accordance with the great Nebular Theory is, I believe, now very generally admitted. But the only authority I shall cite in illustration of this final statement is the Lady Psyche, who commences her exquisite address to her “patient range of pupils” with the words:— “This world was once a fluid haze of light, Till toward the centre set the starry tides, And eddied into suns, that wheeling, cast The planets;” ------------------------------------------------------------------------ APPENDICES. ------- I.—ON THE HEAT GIVEN OUT IN THE CONTRACTION OF THE NEBULA. § 1. FUNDAMENTAL THEOREMS IN THE ATTRACTION OF GRAVITATION. The first theorem to be proved is as follows:— _The attraction of a thin homogeneous spherical shell on any point in its interior vanishes._ [Illustration: Fig. 59.] Take any point P within the sphere. Let this be the vertex of a cone produced both ways, but with a very small vertical angle, so that the small areas S and S´, in which the two parts of the cone cut the sphere, may be regarded as planes. Draw the tangent planes at S and S´. Let the plane of the paper pass through P and be perpendicular to both these tangent planes. Let O P O´ be one of the generators of the cone, and let fall P Q perpendicular to the tangent plane at O, and P Q´ perpendicular to the tangent plane at O´. The volume of the cone with the vertex at P and the base S is ⅓ P Q × S, and the other part of the cone has the volume ⅓ P Q´ × S´. As the vertical angles of the cones are small, their volumes will, in the limit, be in the ratio of O P^3 to O´ P^3, and accordingly ⅓ P Q · S ÷ ⅓ P Q´ · S´ = P O^3 ÷ O´ P^3. But from the figure P Q ÷ P Q´ = P O ÷ P O´, and hence S ÷ O P^2 = S´ ÷ O´ P^2. As the shell is uniform, the masses of the parts cut out by the cones are respectively proportional to S and S´. Hence we see that the attractions of S and S´ on P will neutralise. The same must be true for every such cone through P, and accordingly the total attraction of the shell on a particle inside is zero. The second fundamental theorem is as follows:—