Suppose now that A, B, C, ... are the stations of a network of triangulation projected on or lying on a spheroid of semiaxis major and eccentricity a, e, this spheroid having its axis parallel to the axis of rotation of the earth, and its surface coinciding with the mathematical surface of the earth at A. Then basing the calculations on the observed elements at A, the calculated latitudes, longitudes and directions of the meridian at the other points will be the true latitudes, &c., of the points as projected on the spheroid. On comparing these geodetic elements with the corresponding astronomical determinations, there will appear a system of differences which represent the inclinations, at the various points, of the actual irregular surface to the surface of the spheroid of reference. These differences will suggest two things,--first, that we may improve the agreement of the two surfaces, by not restricting the spheroid of reference by the condition of making its surface coincide with the mathematical surface of the earth at A; and secondly, by altering the form and dimensions of the spheroid. With respect to the first circumstance, we may allow the spheroid two degrees of freedom, that is, the normals of the surfaces at A may be allowed to separate a small quantity, compounded of a meridional difference and a difference perpendicular to the same. Let the spheroid be so placed that its normal at A lies to the north of the normal to the earth's surface by the small quantity [xi] and to the east by the quantity [eta]. Then in starting the calculation of geodetic latitudes, longitudes and azimuths from A, we must take, not the observed elements [phi], [alpha], but for [phi], [phi] + [xi], and for [alpha], [alpha] + [eta] tan [phi], and zero longitude must be replaced by [eta] sec [phi]. At the same time suppose the elements of the spheroid to be altered from a, e to a + da, e + de. Confining our attention at first to the two points A, B, let ([phi]'), ([alpha]'), ([omega]) be the numerical elements at B as obtained in the first calculation, viz. before the shifting and alteration of the spheroid; they will now take the form ([phi]') + f[xi] + g[eta] + hda + kde, ([alpha]') + f'[xi] + g'[eta] + h'da + k'de, [omega] + f"[xi] + g"[eta] + h"da + k"de, where the coefficients f, g, ... &c. can be numerically calculated. Now these elements, corresponding to the projection of B on the spheroid of reference, must be equal severally to the astronomically determined elements at B, corrected for the inclination of the surfaces there. If [xi]', [eta]' be the components of the inclination at that point, then we have [xi]' = ([phi]') - [phi]' + f[xi] + g[eta] + hda + kde, [eta]' tan [phi]' = ([alpha]') - [alpha]' + f'[xi] + g'[eta] + h'da + k'de, [eta]' sec [phi]' = ([omega]) - [omega] + f"[xi] + g"[eta] + h"da + k"de,