Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/44

earth is sometimes taken as a unit, but in the dynamical theory of astronomy the unit of mass is deduced from the units of time and length, combined with the fact of universal gravitation. The astronomical unit of mass is that mass which attracts another body placed at the unit of distance so as to produce in that body the unit of acceleration.In framing a universal system of units we may either deduce the unit of mass in this way from those of length and time already defined, and this we can do to a rough approximation in the present state of science; or, if we expect^[1] soon to be able to determine the mass of a single molecule of a standard substance, we may wait for this determination before fixing a universal standard of mass.

We shall denote the concrete unit of mass by the symbol $[M]$ in treating of the dimensions of other units. The unit of mass will be taken as one of the three fundamental units. When, as in the French system, a particular substance, water, is taken as a standard of density, then the unit of mass is no longer independent, but varies as the unit of volume, or as $[L^{3}]$.

If, as in the astronomical system, the unit of mass is defined with respect to its attractive power, the dimensions of $[M]$ are $[L^{3}T^{-2}]$.

For the acceleration due to the attraction of a mass $m$ at a distance $r$ is by the Newtonian Law ${\frac {m}{r^{2}}}$. Suppose this attraction to act for a very small time $t$ on a body originally at rest, and to cause it to describe a space $s$, then by the formula of Galileo,

$s={\frac {1}{2}}ft^{2}={\frac {1}{2}}{\frac {m}{r^{2}}}t^{2};$

whence $m=2{\frac {r^{2}s}{t^{2}}}$. Since $r$ and $s$ are both lengths, and $t$ is a time, this equation cannot be true unless the dimensions of $m$ are $[L^{3}T^{-2}]$. The same can be shewn from any astronomical equation in which the mass of a body appears in some but not in all of the terms ^[2].

1. See Prof. J. Loschmidt, 'Zur Grösse der Luftmolecule', Academy of Vienna, Oct. 12, 1865; G. J. Stoney on 'The Internal Motions of Gases', Phil. Mag., Aug. 1868; and Sir W. Thomson on 'The Size of Atoms', Nature, March 31, 1870.
2. If a foot and a second are taken as units, the astronomical unit of mass would be about $932,000,000$ pounds.