The Earth's Beginning

by Robert S. Ball

§ 9. A Geometrical Proposition.

The following theorem in elementary geometry will be required:—

Fig. 61.—A Useful Geometrical Proposition.

Let A B and A C be adjacent sides of a parallelogram, Fig. [61], of which A D is the diagonal, and let O be any point in its plane. Then the area O A C is the difference of the areas O A D and O A B.

Draw D Q and C P parallel to O A. Then O A D = O A Q, whence O A D – O A B = O B Q = O A P = O A C.

§ 10. Relation Between the Change of Moment of Momentum and the Force Acting on the Particle.

Fig. 62.—Acceleration of Moment of Momentum equals Moment of Force.

Let A₁ and A₂, Fig. [62], be two adjacent points on the path of the particle, and let A₁ Q and A₂ R be the tangents at those points. Let S Q represent the velocity of the particle at A₁, and SR the velocity of the particle at A₂. Then Q R represents both in magnitude and direction the change in velocity due to the force F, which we suppose constant both in magnitude and direction, while the particle moves from A₁ to A₂ in the small time t; we have also Q R = F t ÷ m.