The acceleration of a falling body is naturally attributed to the presence of the earth; and, though the body approaches the earth in the course of its fall, it is easily recognized that the conditions under which it moves are only very slightly affected by this approach.
Then assuming that the acceleration of one point of a particuar link of the mechanism is known together with the corresponding configuration of the mechanism, the two Vectors Ac and ct can be drawn.
This gives the acceleration of m as modified by the inertia of the wheel.
Now, if m represent the mass of the body in grammes its weight will be mg dynes, for it will require a force of mg dynes to produce in it the acceleration denoted by g.
That at any point the tangent to the hodograph is parallel to the direction, and the velocity in the hodograph equal to the magnitude of the resultant acceleration at the corresponding point of the orbit.
The year 1787 was rendered further memorable by Laplace's announcement on the 19th of November (Memoirs, 1786), of the dependence of lunar acceleration upon the secular changes in the eccentricity of the earth's orbit.
The acceleration of a second point, usually taken at the centre of the crosshead pin, can be found by the principles of 82, but several special geometrical constructions have been devised for this purpose, notably the construction of Klein,i discovered also independently by Kirsch.i But probably the most convenient is the construction due to G.
Sometimes, as on the Central London railway, the acceleration of gravity is also utilized; the different stations stand, as it were, on the top of a hill, so that outgoing trains are aided at the start by having a slope to run down, while incoming ones are checked by the rising gradient they encounter.
If a is variable, then the formula must be applied in a series of steps, each step corresponding to a time interval over which the acceleration may be assumed uniform.
In this expression it is assumed that the acceleration is uniform, and this assumption is sufficiently accurate for any practical purpose to which the above formula would be applieu in the ordinary working of a locomotive.