The analytic method sought to express the moon's motion by integrating the differential equations of the dynamical theory.

The mean brightness varies as z3 (or as r3), and the integral found by multiplying it by zdz and integrating between o and co converges.

After the establishment of universal gravitation as the primary law of the celestial motions, the problem was reduced to that of integrating the differential equations of the moon's motion, and testing the completeness of the results by comparison with observation.

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Apparatus is added to some dynamometers by means of which a curve showing the variations of P on a distance base is drawn automatically, the area of the diagram representing the work done; with others, integrating apparatus is combined, from which the work done during a given interval may be read off directly.

A recording drum or integrating apparatus may be arranged on the pulley frames.

Only by taking infinitesimally small units for observation (the differential of history, that is, the individual tendencies of men) and attaining to the art of integrating them (that is, finding the sum of these infinitesimals) can we hope to arrive at the laws of history.

The Mean Specific Heat, Over Any Range Of Temperature, May Be Obtained By Integrating The Formulae Between The Limits Required, Or By Taking The Difference Of The Corresponding Values Of The Total Heat H, And Dividing By The Range Of Temperature.