G03 each in mean latitudes, and near each other, say about five degrees of latitude apart, the probable error of the resulting value of the ellipticity will be approximately 5-^, e being expressed in seconds, so that if e be so great as 2" the probable error of the resulting ellipticity will be greater than the ellipticity itself. It is not only interesting, but necessary at times, to calculate the attraction of a mountain, and the consequent disturbance of the astronomical zenith, at any point within its influence. The deflection of the plumb-line, caused by a local attraction whose amount is AS, is measured by the ratio of AS to the force of gravity at the station. Expressed in seconds, the deflection A is A =12". 447. -A, P where p is the mean density of the earth, 8 that of the attracting mass, the linear unit in expressing A >eing a mile. Suppose, for instance, a table-land whose form is a rectangle of twelve miles by eight miles, having a height of 500 feet and density half that of the earth ; let the observer be two miles distant from the middle point of the longer side. The deflection then is 1" 472 ; but at one mile it increases to 2" 20. At sixteen astronomical stations in the English Survey the disturbance of latitude due to the form of the ground has been computed, and the following will give an idea of the results. At six stations the deflection is under 2", at six others it is between 2" and 4", and at four stations it exceeds 4". There is one very exceptional station on the north coast of Banffshire, near the village of Portsoy, at which the deflection amounts to 10", so that if that village were placed on a map in a position to correspond with its astronomical latitude, it would be 1000 feet out of position ! There is the sea to the north and an undulating country to the south, which, however, to a spectator at the station does not suggest any great disturbance of gravity. A somewhat rough estimate of the local attraction from external causes gives a maximum limit of 5", therefore we have 5" unaccounted for, or rather which must arise from unequal density in the underlying strata in the surrounding country. In order to throw light on this remarkable phenomenon, the latitudes of a number of stations between Nairn on the west, Fraserburgh on the east, and the Grampians on the south, were observed, and the local deflections determined. It is somewhat singular that the deflections diminish in all directions, not very regularly certainly, and most slowly in a south-west direc tion, finally disappearing, and leaving the maximum at the original station at Portsoy. The method employed by Dr Hutton for computing the attraction of masses of ground is so simple and effectual that it can hardly be improved on. Let a horizontal plane pass through the given station; let r, be the polar co ordinates of any point in this plane, and r, 0, z, the co ordinates of a particle of the attracting mass ; and let it be required to find the attraction of a portion of the mass contained between the horizontal planes 2 = 0, z = h, the cylindrical surfaces r = r l ,r = r< n and the vertical planes = ly 6 = $ 2 . The component of the attraction at the station or origin along the line = is " h r 9 cos QdrdQdz = Sh (sin 0> - sin ej log y a + S r 1 + ( By taking r 2 - r x sufficiently small, and supposing h also small, as it usually is, compared with i + r z , the attraction is =- 8(r 2 - r 1 )(sin 2 - sin X ) - , where r=^(r l + r. 2 ). This form suggests the following pro cedure. Draw on the contoured map a series of equidistant circles, concentric with the station, intersected by radial lines so disposed that the sines of their azimuths are in arithmetical progression. Then, having estimated from the map the mean heights of the various compartments, the cal culation is obvious. In mountainous countries, as near the Alps and in the Caucasus, deflections have been observed to the amount of as much as 29". On the other hand, deflections have been observed in flat countries, such as that noted by Professor Schweitzer, who has shown that, at certain stations in the vicinity of Moscow, within a distance of 16 miles the plumb- line varies 16" in such a manner as to indicate a vast deficiency of matter in the underlying strata. But these are exceptional cases. 1 Since the attraction of a mountain mass is expressed as a numerical multiple of S : p, the ratio of the density of the mountain to that of the earth, if we have any independent means of ascertaining the amount of the deflection, we have at once the ratio p : S, and thus we obtain the mean density of the earth, as, for instance, at Schiehallion, and more recently at Arthur s Seat. A com pact mass of great density at a small distance under the surface of the earth will produce an elevation of the mathe matical surface which is expressed by the formula
- . ,-iL
where a is the radius of the (spherical) earth, (1 - ) the distance of the disturbing mass below the surface, fj. the ratio of the disturbing mass to the mass of the earth, and aO the distance of any point on the surface from that point, say Q, which is vertically over the disturbing mass. The maximum value of y is at Q, where it is ft The deflection at the distance aO is fik sin 6 A = (H-P-2/tcos 6) or since is small, putting h + k= 1, The maximum deflection takes place at a point whose distance from Q is to the depth of the mass as 1 : *j 2, and its amount is J_ VL 3V3W If, for instance, the disturbing mass were a sphere a mile in diameter, the excess of its density above that of the surrounding country being equal to half the density of the earth, and the depth of its centre half a mile, the greatest deflection would be 5", and the greatest value of y only two inches. Thus a large disturbance of gravity may arise from an irregularity in the mathematical surface whose actual magnitude, as regards height at least, is extremely small. The effect of the disturbing mass fj. on the vibrations of a pendulum would be a maximum at Q ; if v be the number of seconds of time gained per diem by the pendulum at Q, and cr the number of seconds of angle in the maximum deflection, then it may be shown that _7rV3 !? nr 1 In the Philosophical Transactions for 1855 and 1859 will be found Archdeacon Pratt s calculations of the attractions of the Hima layas and the mountain region beyond them, and the consequent deflection of the plumb-line at various stations in India ; the subject, which presents many anomalies and difficulties, is very fully gone into in his treatise on the figure of the earth. His computed deflections
are vastly greater than anything brought to light by observation.
