On the Constant P in Observations of Terrestrial Magnetism

Abstract

I REGRET that Prof. Rücker should have largely misunderstood my last letter. I have not raised the question of fallible observations at all. Referring to the correspondence on pages 127–8 of the present volume of NATURE, my principal contention was and is that the ordinarily accepted formula for P differs by terms of the second and higher orders from Gauss's theory, and that that difference necessarily persists in any rigorous expansion of the formula. By the ordinarily accepted formula for P I mean Prof. Rücker's formula (α); and by Gauss's theory I mean my formulæ (1), (2), and (3). From two observations of f(u), made respectively at the distances r and r₁, the L of Gauss's theory might be found by a direct solution of equations (1) and (2); but instead of that, it is customary to find L from equations (7) and (8) by substituting in them the value of P₀ computed through equation (α). To render the latter procedure rigorous, P should be used in (7), and P₁ in (8). Equation (11) shows that P and P₁ differ by quantities of the second and higher orders, and as the ordinarily accepted value of P₀ lies between P and P₁, it necessarily differs from one or both of these quantities, and therefore from Gauss's theory, by terms of the second and higher orders.