Page:Encyclopædia Britannica, Ninth Edition, v. 10.djvu/182

Rh 1 70 Here n is the normal or radius of curvature perpendicular I to the meridian ; both n and p correspond to latitude gbl, I and po to latitude ,‘,(¢+§b'). For calculations of latitude. and longitude, tables of the logaritlunic values of p sin 1", 92 sin 1", and :'.’np sin 1" are necessary. The following table contains these logarithms for every ten 1nin11tes of latitude from 52' to 53° computed with the elements a = 20926060 and a :c=295 : 294 :— 1 1 Lat’ l Log’ sin 1" ' Lag’ 1) sin 1" I mg' ’ I 52 0 79939434 7'99‘2.8231 ' 037131 10 9309 8190 | 29 20 J 9135 81 ‘.28 30 9060 S10: 26 40 8936 8065 24 50 8812 8024 ‘.23 53 0 8683 79:2 2'2 The logaritlnn in the last column is that required also for the calculation of spherical excesses, the spherical excess of a triangle ab sin C 1 ' .— » v j_ . iemg expressed by gpn sin 1,, It is frequently necessary to obtain the coordinates of one point with reference to another point ; that is, let a perpendicular are be drawn from B to the meridian of A meeting it in P, then, a being the azimuth of B at A, the coordinates of B with reference to A are AP=s cos (a - §-e), BP=s sin (a—— ﬁe), where e is the spherical excess of APB, viz., s‘-’ sin acos a multiplied by the quantity whose logarithm is in the fourth column of the above table. I rre_r/ul(u'z'tz'es qf the Earth’s Szu;/"ace. In considering the effect of unequal distribution of matter in the earth’s crust ou the form of the surface, we may simplify the matter by disregarding the considerations of rotation and excentricity. In the ﬁrst place, supposing the earth a sphere covered with a ﬁlm of water, let the density p be a function of the distance from the centre so that surfaces of equal density are concentric spheres. Let now a disturbance of the arrangement of nutter take place, so that the density is no longer to be expressed by p, a function of 7° only, but is expressed by p + p’, where p’ is a function of three coordinates 0, (,b, 9'. Then p’ is the density of what may be designated disturbing matter ; it is positive in some places and negative in others, and the whole quan- tity of matter whose density is p’ is zero. The previously spherical surface of the sea of radius (L now takes a new form. Let P be a point on the disturbed surface, P’ the corresponding point vertically below it on the undisturbed surface, 1’P’=u. The knowledge of it over the whole sur- face gives us the form of the disturbed or actual surface of the sea ; it is an equipotential surface, and if V be the potential at P of the disturbing matter p’, M the mass of the earth, M a+u ‘I M +V=C-=5,; ——,;,u+V. As far as we know, u is always a very small quantity, and we have with sufﬁcient approximation u = 4%, where 8 is ma the mean density of the earth. Thus we have the disturb- ance in elevation of the sea-level expressed in terms of the potential of the disturbing matter. If at any point P the value of it remain constant when we pass to any adjacent point, then the actual surface is there parallel to the ideal spherical surface; as a rule, however, the normal at P is in- clined to that at P’, and astronomical observations have GEODESY shown that this inclination, amounting ordinarily to one or two seconds, may in some cases exceed 10, or, as at the foot of the Himalayas, even 30 seconds. By the expression “mathematical figure of the earth” we mean the surface of the sea produced in imagination so as to percolate the con- tinents. We see then that the effect of the uneven distri- bution of matter in the crust of the earth is to produce small elevations and depressions on the mathematical sur- face which would be otherwise spheroidal. No geodesist can proceed far in his work without encountering the irre- gularities of the mathematical surface, and it is necessary that he know how they affect his astronomical observations. The whole of this subject is dealt with in his usual elegant manner by Bessel in the Astr-onrmuls-c/ac i'cu-7z.~'z'«.-/elm, Nos. 3:29, 330, 331, in a paper entitled “Ueber den I-Iinﬂuss der Unregelmiissigkeiten der Figur der Erde auf gcodii- tisehe Arbeiten, &c.” But without entering into further details it is not difficult to see how local attraction at any station affects the determinations of latitude, longitude, and azimuth there. Let there be at the station an attraction to the north—east throwing the zenith to the south-west, so that it takes in the celestial sphere a position Z’, its undisturbed position being Z. Let the rectangular components of the displace- ment ZZ' be 5 measured southwards and 1] measured west- wards. Now the great circle joining Z’ with the pole of the heavens P makes there an angle with the meridian PZ=-q cosec PZ'=7; see (,b, where gb is the latitude of the station. Also this great circle meets the horizon in a point whose distance from the great circle PZ is 'qSeC(,bsiI1<;b ='qtal1§b. That is, a meridian mark, ﬁxed by observa- tions of the pole star, will be placed that amount to the east of north. Hence the observed latitude requires the correc- tion 5; the observed longitude a correction 77 see (,b; and any observed azimuth a correction vytangb. Here it is supposed that azimuths are measured from north by east, and longitudes eastwards. The expression given for it enables one to form an ap- proximate estimate of the effect of a compact mountain in raising the sea-level. Take, for instance, Ben Nevis, which contains about a couple of cubic miles 3 a simple calculation shows that the elevation produced would only amount to about 3 inches. In the ease of a mountain mass like the Himalayas, stretching over some 1500 miles of country with a breadth of 300 and an average height of 3 miles, although it is dilﬁcult or impossible to find an expression for V, yet we may ascertain that an elevation amounting to several hundred feet may exist near their base. The geodetic-al operations, however, rather negative this idea, for it is shown in a paper in the I’/u'losoplt2'r-((1 .i[I([/((2‘l')l€ for August 1878 by Colonel Clarke that the form of the sea—level along the Indian arc departs but slightly from that of the mean ﬁgure of the earth. If this be so, the action of the Himalayas must be counteracted by subterranean tenuity. Suppose now that A, B, C,. . . . are the stations of a net_.- work of triangulation projected on or lying on a spheroid of semiaxis major a.nd excentricity a, c, 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, lougitudes, 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 correspond- ing astronomical determinations, there will appear a system of differences which represent the inclinations, at the vari- ous 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 refer-