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.
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 P3 to O´ P3, and accordingly ⅓ P Q · S ÷ ⅓ P Q´ · S´ = P O3 ÷ O´ P3. But from the figure P Q ÷ P Q´ = P O ÷ P O´, and hence S ÷ O P2 = S´ ÷ O´ P2.
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:—