ence 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, com- pounded of a meridional difference and a difference per- pendicular 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 ξ and to the east by the quantity η Then in starting the calculation of geodetic latitudes, longitudes, and azimuths from A, we must take, not the observed elements :φ α, but for φ, φ+ξ, and for α, α + η; tan φ, and zero longit11de 1n11st be replaced by η sec φ. At the same time suppose the elements of the spheroid to be altered from a, e to (t+du, c+clc. Confining our attention at first to the two points A, B, lct (¢>'), ((1'), (to) be the numerical elements at 1} as obtained in the first calcu- la.tion, viz., before the shifting and alteration of the spheroid ; they will now take the form
where the coefiicientsf, _r/, . . . &c. can be numerically calcu- lated. 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 5', 27' be the components of the inclination at that point, then we have g'=(¢’)—¢’+ f.§+ _r/17+ Iula+I.-dc, 7;’ tan ¢’=- (a') — a’ +f '§+ _r/17+ h’¢la+I/clc, 1;’ sec ¢'= (co) — w+f'§+g"1;+h"(l(L+k"¢lc, where <,f)', a’, (J) are the observed elements at B. Here it appears that the observation of longitude gives no ad- (litional information, but is available as a check upon the azimuthal observations. If now there be a number of astronomical stations in the triangulation, and we form equations such as the above for each point, then we can from them determine those values of 5, 27, (la, «Iv, which make the quantity 5‘-’ + 27'-’+ 59+ q7"3+ a minimum. Thus we obtain that spheroid which best represents the surface covered by the triangulation. In the Jccozmt qf I/zc I’rz'm:z'pal Tri(m__r/u/uz‘z'on of Greru‘ ];'ritm'n (md Irclruul will be found the (letermination, from 75 equations, of the spheroid best representing the surface of the British Isles. Its elements are a= 20927005 =e 295 feet, I) : (1 - b: 980 =e 8 ; and it is so placed that at Green- wich Observatory S = l"°864, 27 = — O""546. Taking Durham Observatory as the origin, and the tan- gent plane to the surface (determined by 5: — 0"'664, 27= — 4”'1l7) as the plane of .2: and 3/, the former measured northwards, and 2 measured vertically downwards, the equation to the surface is "995-24953.:-9+ '99288005]/2 + '9976-3052:” — 0'00671003.‘r_'z — 41655070z=O. AItz't2u.hs. The precise determination of the altitude of his station is a matter of secondary importance to the geodesist; never- theless _it is usual to observe the zenith distances of all trigo- nometrical points. The height of a station does indeed influence the observation of terrestrial angles, for a vertical line at B does not lie generally in the vertical plane of A, but the error (which is very easily investigated) involved in the neglect of this consideration is much smaller than the errors of observation. Again, in rising to the l1cigl1t it above the surface, the centrifugal force is increased and the magnitude and direction of the attraction of the l 71 earth are altered, and the effect upon the observation of latitude is a very small error expressed by the formula h g’ —g (7 ' T the equator and at the pole. This is also a quantity which may be neglected, since for ordinary mountain heights it amounts to only a few hundreths of a second. The uncertainties of terrestrial refraction render it im- possible to determine accurately by vertical angles the heights of distant points. Generally speaking, refraction is greatest at about daybreak ; from that time it diminishes, being at a minimum for a couple of hours before and after mid-day ; later in the afternoon it again increases. This at least is the general march of the phenomenon, but it is by no means regular. The vertical angles measured at the station on Hart Fell showed on one occasion in the month of September a refraction of double the average amount, lasting from 1 RM. to 5 RM. The mean value of the co- efiicient of refraction is determined from a very large num- ber of observations of terrestrial zenith distances in Great Britain is '0792=& ‘0047; and if we separate those rays which for a considerable portion of their length cross the sea from those which do not, the former give It-= 0813 and the latter It-=°O753. These values are determined from high stations and long distances; when the distance is short, and the rays graze the ground, the amount of refrac- tion is extremely uncertain and variable. A case is noted in the Indian Survey where the zenith distance of a station 101') miles off varied from a depression of 4' 52"'6 at 4.30 P.1I. to an elevation of 2’ i‘4”'0 at 10.50 P.M. If la, It’ be the heights above the level of the sea of two stations, 90° + 8, 90° + 3' their mutual zenith distances (8 being that observed at It), s their distance apart, the earth being regarded as a sphere of radius=a, then, with sufficient precision, h'— h=s tan (sl_ 2Z"‘— 8) , "(L It-—h'=s tan (.91 _ 2k— 8’) . ‘.2.rL If from a station whose height is /L the horizon of the sea be observed to have a zenith distance 90° + 8, then the above formula gives for It the value sin f2¢>,' where g, g’ are the values of gravity at h=gt_ _ tan‘~’8 . 2 1 — 2].: Suppose the depression 8 to be 7L minutes, then It = 1'05-tn? if the ray be for the greater part of its length cross- ing the sea ; if otherwise, fl. = 1'040n‘3. To take an example : the mean of eight observations of the zenith distance of the sea horizon at the top of Ben Nevis is 91° 4' 48", or E3: 64'S; the ray is pretty equally disposed overland and water, and hence it = 1'047n'~’= 4396 feet. The actual height of the hill by spirit-levelling is 44-06 feet, so that the error of the height thus obtained is only 10 feet. Longitucle. The determination of the difference of longitude between two stations A and B resolves itself into the determination of the local time at each of the stations, and the compari- son by signals of the clocks at A and B. Whenever tele- graphic lines are available these comparisons are made by electro-telegraphy. A small and delicately-made apparatus introduced into the mechanism of an astronomical clock or chronometer breaks or closes by the action of the clock a galvanic circuit every second. I11 order to record the minutes as well as seconds, one second in each minute, namely that numbered 0 or 60, is omitted. The seconds are recorded on a chronograph, which consists of acylinder re-
volving uniformly at the rate of one revolution per minute