MAY 19, 1923 MATHEWS: PHYSICAL DIMENSIONS 203 
This may be expected to involve both the specific inductive capacity 
and magnetic permeability. To get its dimensions we have to turn 
to the Newtonian law of gravitation. 
MM’ g 
Equation (17), which states that the force of attraction is directly 
as the product of the masses and inversely as the square of the distance, 
leaves out the gravitational permeability and accordingly does not 
balance dimensionally. The equation is ordinarily made to balance 
3 5 
dimensionally by arbitrarily ascribing to g, the dimensions i 
(17) 
But this is incorrect, for g is a numerical constant without dimensions, 
having the value of 6.66 x 10-8 and is in the equation because the 
unit of mass has been arbitrarily chosen as that of 1 cc. of water at a 
certain temperature. Had unit mass been defined as that mass 
exerting unit force on another similar mass at unit distance, g, would 
have had the value of unity and would not have been in. It is clear 
that the factor G, gravitational permeability, must be put into the 
denominator. Hence the equation would be: 
MM’g 
ig ~ Gd 
| and G will have the dimensions 
M T? a2 
(19) (G) = (47) 2 (uT?) = eKy = L? X T? 
That is, gravitational permeability is equal to magnetic permeability 
X specific inductive capacity x electric quantity or area. Its final di- 
mension is (T?), which is either the square of a period, or the reciprocal 
of angular acceleration. The value of G will probably be unity for all 
_ substances except possibly hydrogen, for hydrogen alone is exceptional 
in its atomic weight. All other elements have atomic weights which are 
whole numbers, the weight of oxygen being taken as 16. Hydrogen 
alone has an atomic weight which is not a whole number. Its atom 
instead of a weight of 1, is 1.008. It is not probable that this is due 
to the presence of isotopes. It may be attributed to the fact that the 
gravitational permeability of hydrogen is less than unity and probably 
1 ‘ 
in the proportion (cms Harkins has ascribed the difference 
’ See for example Planck. Heat Radiation p. 174-175. (Translation by Masius.) 
