Volume 6 (1960-1961) Issue 3 Pages 75-77
The inverse problem of this title, formula for calculating meridian-length (M) from equator to a point when its latitude (ψ) is given, is already well-known to us. For example, the expansion of the exact elliptic integral formula in series in terms of powers of e2 till e10 is given in Jordan-Eggert: Handbuch der Vermessungskunde III-1, to which the equations (1), (2) and (3) in my short paper are referred. I have not ever seen the formula for the above title . I suppose, this is due to the following two reasons : first, once the table for calculating M from ψ is made, ψ is easily found by inversely interpolating it; second, the formula of the title is not so simple as its inverse formula (2). However, as I do not think it worthless to introduce the formula of the title, I tried to do it. A series in terms of powers of e2 till e8 is obtained by inversely expanding after the method of successive approximation, (equations (5), (6) and (7)). They are apparently intricate as compared with the inverse formula, because our exact formula is not expressed as a simple integral function of M in itself . The higher are powers of e2, the more rapidly increases the number of harmonics contained in them . The term of e8 has 14 harmonics, but only 5 harmonics in the inverse problem. The effect of the term of e8 is, however, very small over a quarter of the surrounding circle of the ellipsoid. Even if we neglect the term of e8, we can find sufficiently correct value of ψ till 10-4 second . When we do so, the equation (6) is shortened about half and becomes a little easy for use.