§ 48. Allowance must be made for the difference in the epochs, and for the fact that the number of constants assumed to be worth retaining was different in each case. Gauss, for instance, assumed 24 constants sufficient, whilst in obtaining the results given in the table J. C. Adams retained 48. Some idea of the uncertainty thus arising may be derived from the fact that when Adams assumed 24 constants sufficient, he got instead of the values in the table the following:—
| g10 | g11 | h11 | |
| 1842-1845 | +.32173 | +.02833 | −.05820 |
| 1880 | +.31611 | +.02470 | −.06071 |
Some of the higher constants were relatively much more affected. Thus, on the hypotheses of 48 and of 24 constants respectively, the values obtained for g20 in 1842-1845 were -.00127 and -.00057, and those obtained for h31 in 1880 were +.00748 and +.00573. It must also be remembered that these values assume that the series in positive powers of r, with coefficients having negative suffixes, is absolutely non-existent. If this be not assumed, then in any equation determing X or Y, gmn must be replaced by gmn + gm−n, and in any equation determining Z by gmn − {n/(n + 1)} gmn; similar remarks apply to hmn and hm−n. It is thus theoretically possible to check the truth of the assumption that the positive power series is non-existent by comparing the values obtained for gmn and hmn from the X and Y or from the Z equations, when gm−n and hm−n are assumed zero. If the values so found differ, values can be found for gm−n and hm−n which will harmonize the two sets of equations. Adams gives the values obtained from the X, Y and the Z equations separately for the Gaussian constants. The following are examples of the values thence deducible for the coefficients of the positive power series:—
| g−10 | g−11 | h−11 | g−40 | g−50 | g−60 | |
| 1842-1845 | +.0018 | −.0002 | −.0014 | +.0064 | +.0072 | +.0124 |
| 1880 | −.0002 | −.0012 | +.0015 | −.0043 | −.0021 | −.0013 |
Compared to g40, g50 and g60 the values here found for g−40, g−50 and g−60 are far from insignificant, and there would be no excuse for neglecting them if the observational data were sufficient and reliable. But two outstanding features claim attention, first the smallness of g−10, g−11 and h−11, the coefficients least likely to be affected by observational deficiencies, and secondly the striking dissimilarity between the values obtained for the two epochs. The conclusion to which these and other facts point is that observational deficiencies, even up to the present date, are such that no certain conclusion can be drawn as to the existence or non-existence of the positive power series. It is also to be feared that considerable uncertainties enter into the values of most of the Gaussian constants, at least those of the higher orders. The introduction of the positive power series necessarily improves the agreement between observed and calculated values of the force, but it is more likely than not to be disadvantageous physically, if the differences between observed values and those calculated from the negative power series alone arise in large measure from observational deficiencies.
Table XLV.—Axis and Moment of First Order Gaussian Coefficients.
| Epoch. | Authority for Constants. | North Latitude. | West Longitude. | M/R3 in G.C.S. units. |
| ° ′ | ° ′ | |||
| 1650 | H. Fritsche | 82 50 | 42 55 | .3260 |
| 1836 | ” | 78 27 | 63 35 | .3262 |
| 1845 | J. C. Adams | 78 44 | 64 20 | .3282 |
| 1880 | ” | 78 24 | 68 4 | .3234 |
| 1885 | Neumayer-Petersen and Bauer | 78 3 | 67 3 | .3224 |
| 1885 | Neumayer, Schmidt | 78 34 | 68 31 | .3230 |
§ 49. The first order Gaussian constants have a simple physical meaning. The terms containing them represent the potential arising from the uniform magnetization of a sphere parallel to a fixed axis, the moment M of the spherical magnet being given by
M = R3 { (g10)2 + (g11)2 + (h11)2}1/2,
where R is the earth’s radius. The position of the north end of the axis of this uniform magnetization and the values of M/R3, derived from the more important determinations of the Gaussian constants, are given in Table XLV. The data for 1650 are of somewhat doubtful value. If they were as reliable as the others, one would feel greater confidence in the reality of the apparent movement of the north end of the axis from east to west. The table also suggests a slight diminution in M since 1845, but it is open to doubt whether the apparent change exceeds the probable error in the calculated values. It should be carefully noticed that the data in the table apply only to the first order Gaussian terms, and so only to a portion of the earth’s magnetization, and that the Gaussian constants have been calculated on the assumption that the negative power series alone exists. The field answering to the first order terms—or what Bauer has called the normal field—constitutes much the most important part of the whole magnetization. Still what remains is very far from negligible, save for rough calculations. It is in fact one of the weak points in the Gaussian analysis that when one wishes to represent the observed facts with high accuracy one is obliged to retain so many terms that calculation becomes burdensome.