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*ach in mem latitudes, and near each otter, say about five degrees of latitude apart, the probable error of the resulting value of the ellipticity will be approximately * ,4^ < being expressed in seconds, so that if < 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 = W.U7.-X ,

where p is tho mean density of the earth, 8 that of the attracting mass,—the linear unit in expressing A being 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 l"-472; but at one mile it increases to 2"°20. At sixteen astronomical stations in tho English Surrey 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 i". 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 wonld be 1000 feet out of position! There is the sea to the north and an undulating country to tho 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, tho 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 sonth-west direction, 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, 6 be the polar coordinates of any point in this plane, and r, $, e, the coordinates of a particle of the attracting moss; and let it be required to find the attraction of a portion of the mass contained between the horizontal planes z = 0, i h, the cylindrical surfaces r*>rv r-r,, and the vertical planes 6 6y The component of the attraction at the station or origin along the line 0—0 if

. /•r*/*B'/'> rt 008 Mrdedz

J J J ff*+l«)l


By taking rt - r. sufficiently small, and supposing h also small, as it usually is, compared with r1 + rv the attraction is

where r- Jfo + r,). This form suggests the following pro

cedure. Draw on tho contoured map a scries of equidistant circles, concentric with the station, intersected by radial lines so disposed that tho sines of thoir azimuths are in arithmetical progression. Then, having estimated from the map the mean heights of the various compartments, the calculation 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 tho vicinity of Moscow, within a distance of 10 miles the plumbline 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 muss is expressed as a numerical multiple of S :p, tho ratio of tho density of the mountain to that of tho earth, if we have any independent means of ascertaining the amount of tho deflection, we have at once the ratio p : S, and thus «e obtain the mean density of the earth, as, for instance, at Schiehallion, and more recently at Arthur's Seat. A compact moss of great density at a small distance under the surface of the earth will produce an elevation of the mathematical surface which is expressed by the formula

y~ai1 i (1 +F^2Feosl)I "1 i'

where a is the radius of the (spherical) earth, a(l-J) the distance of the disturbing moss below the surface, n the ratio of the disturbing mass to the mass of the earth, and a$ 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

The deflection at the distance aO is
A_ gtifal
""(l + i'-atcos*)*
or since $ is small, putting h+k—l,


The maximum deflection takes place at a point wbosa distance from Q is to the depth of the mass as 1: J 2, and its amount is


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 largo 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 p on the vibrations of a pendulum would be a maximum at Q; if y be the number of seconds of time gained per diem by the pendulum at Q, and a the number of seconds of angle in the maximum deflection, then it may be shown that »_»V3 . 10'

1 la tho Philosophical Transactions for J 855 and 1859 will bo found Archdeacon Pratt's calculations of the attraction* of tho Himalayas and the mountain region beyond them, and .the consequent deflection of the plumb-line at various stations in Indian the subject, which presents many anomalies and difficulties, is very fully gone into in his treatise on the figure of the esrth. His computed deflections are vastly greater than anything brought to light by observation.

A101 Alci Alina

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80 that the number of seconds of time by which at the then on expressing x, *', ?, :' in terms of u and x, maximum the pendulum is accelerated is about half the

U=1 -cos u cos u' cos w – sin u sin u'; number of seconds of angle in the maximum deflectiun. also, if y be the third side of a spherical triangle, of shira tio

sides are – u and * -u' and the included angle , nang Principles of Calculation.

subsidiary angle y such that Let a, a' be the mutual azimuths of two points P, Q on a spheroid,

sin y sin = e sin "";cos ""+" , k the chord line joining them, H, M' the angles made by the chord with the normals at Pand Q, ,0', their latitudes and difference of we obtain finally the following equations : longitude, and **"* + -1=0 the equation of the surface ;

k=2a cos e sin
then if the plane zz passes through P the co-ordinates of P and Q
will be

cos u=a sec y sin
as coso,
is cos cos con

cos n'= A' sec y sin
39 = cos o' sin mo,

sin sin a = cos u' sin == 2 c) sin ø, z = 2(2–64) sin d'

sin s' sin a'= cos u sin w. where A=(1 - sin p)!, A'-(1-o sin? 0)*, and e is the eccen. These determine rigorously the distance, and the mutnal zenith tricity. Let f, g, h be the direction cosines of the normal to that distances and azimuths, of any two points on a spheroid whose plane which contains the normal at P and the point Q. and latitudes and difference of longitude are given. whose inclinations to the meridian plane of P is = a; let also By a series of reductions from the cyuations containing ( 5 il 1, m, n and l', m', n' be the direction cosines of the normal at P, may be shown that and of the tangent to the surface at P which lies in the plane passing through Q, then since the first line is perpendicular to each

a+a'-5 + 5 + w($? - o)? cos ‘o, sin po+ ..., of the other two and to the chord k, whose direction cosines are proportional to ' - x, y-y,-2, we have these three equations

There do is the mean of o and , and the higher powers of e are

neglected. A short computation will show that the small quantity Il – 2) + gy' +hz - 2) = 0

on the right-hand side of this equation can never amount even to fl + gm +hn = 0

the ten thousandth part of a second, which is, practically speaking fli + gm' + hn' = 0.

zero; consequently the sum of the azimuths are on the spheroi); Eliminate f, g, h from these equations, and substitut

equal to the sum of the spherical azimuths, whence foDous thig

very important theorem (known as Dalby's theoran). If o, be 1 = cos o l'= - sin o cos a

the latítudes of two points on the surface of a spheroid, 6 their m = 0 m'= sin a

difference of longitude, a, a' their reciprocal azimuths, n = sino n'= cos o cos a, and we get (ad - x) sin o+y' cot a-(4-2) cos o = 0.

tan = cotate'. The substitution of the values of 2, 2, s', y, ' in this equation will give immediately the value of cot a; and if we put , 5' for the corresponding azimuths on a sphere, or on the supposition e=0, The vertical plane at P passing through Q and the vertical plane the following relations exist

at Q passing through P cut the surface of the spheroid in two distinct curves. The greatest distance apart of these curves is, il

ay - the mean azimuth of PQ,
cot a'- cot (= - COS O'Q

cos P, sin 223
cos o A
A sin ¢ - A sin $'= sin w Q.

This is a very small quantity; for eren in the case of a line of 100 Jl from Q we let fall a perpendicular on the meridian plano of P,

miles in length having a mean azimnth a = 45° in the latitude of and from P let fall a perpendicular on the meridian plane of a

Great Britain, it will only amount to half an inch, whilst for a line then the following equations become geometrically evident :

of fifty miles it cannot exceed the sixteenth part of an inch. The

geodesic line joining P and Q lies wholly between these tiro curves, i k sin u sin a = COS sin.

If we designate by P', Q' the two curves (the former being that in the vertical plane through P), then, neglecting quantities of the

order c03, where is the angular distance of P and Q at the centre k sin n' sin d'= cos o sin ».

of the earth, the geodesic curve makes with Pat Pan angle equal

to the angle it makes with Qʻat Q, each of these angles being : Now in any surface = 0 we have

third of the angle of intersection of P' and Q'. The difference of L=12 )* +(^ – 2)*+6 - 2)

length of the geodesic line and either of the curves P", Qis, s being the length of either,

a chi cosa do sino 200 ·
- Cos j =-

At least this is an approximate expression. Supposing the angle
PQ to be as much as 10°, this quantity would be less than oue

hundredth of an inch.
cos M'= -

An idea of the course of a geodesic line may be gathered from the following example. Let the line be that joining

Cadiz and St Petersburg, whose approximate positions are in the present case, if we put


St Petersburg.
Lat. 36° 22' x....

...59° 56' N

Long. 6° 18' w.............30° 17' E. then

If G be the point on the geodesic corresponding to Fon that one of the plane curves which contains the normal at

Cadiz (by “corresponding" we mean that F and G are on a Cos u=EAU; cos w Au.

meridian) then G is to the north of F; at a quarter of the Lot u be such an angle that

whole distance from Cadiz GF is 458 feet, at half the dis(1-r*)t sin o = A sin A sin m

See a paper “On the course of Geodesic Lines on the Eartla's cos y = A cos #

Surfaco" in the Philosophical Magazine for 1870.

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12 - vertet (yo, epitet e audio

u foto - e in cur - vi inte denominado

tance it is 637 feet, and at three quarters it is 473 feet. The azimuth of the geodesic at Cadiz diners 20" from that of the vertical plane, which is tho astronomical azimuth. The azimuth of a geodesic line cannot be observed, so that the line does not enter of necessity into practical geodesy, although many formulas connected with its use are of great simplicity and elegance. The geodesic line has always held a more important place in the science of geodesy among the mathematicians of France, Germany, and Russia than has been assigned to it in the operations of the English and Indian triangulations. Although the observed angles of a trlangulation are not geodesic angles, yet in the calculation of the distance and reciprocal bearings of two points which are far apart, and are connected by a long chain of triangles, we may fall upon the geodesic line in this manner:—

If A, Z be the points, then to start the calculation from A, we obtain by some preliminary calculation the approximate azimuth or Z, or the angle made by the direction of Z with the side AB or AO of the first triangle. Let Pj be the point where this line intersects BC; then, to find P., where the fine cuts the next triangle side CD, we make the angle BPjP, such that BP,P, + BP1A=180°. This fixes Pa, and P. is fixed by a repetition of the same process; eoforP,, P,.... Now it is clear that the points P,, P., Ps so computed are those which would be actually fixed by an observer with a theodolite, proceeding in the following manner. Having set the instrument up at A, and turned the telescope in the direction of the computed bearing, an assistant places a mark Pj on the line BC, adjusting it till bisected by the cross-hairs of the telescope at A. The theodolite is then placed over P,, and the telescope turned to A ; the horizontal circle is then moved through 180°. The assistant then places a mark Ps on the line CD, so as to be bisected by the telescope, which is then moved to Ps, and in the same manner P, is fixed. Now it is clear that the series of points Pj, P9, Pj approaches to the geodesic line, for the plane of any two consecutive elements P»-i P*, P« Pn+i contains the normal at P».

From the formula} which we have given above, expressing the mutual relations of two points P, Q on a spheroid, we may obtain the following solution of the problem: Given the latitude <j> of P, with the azimuth a and distance s of Q, to determine the latitude and longitude of Q and the back azimuth a.

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Now in every measured aro there are not only the extremo stations determined in latitude, but also a number of intermediate stations, so that if there be i+1 stations there will be i equations *,=x1+.Au + 5f1t> + A1 •>«*,+/,«+»,»+»,

Xi = *I+/ii»+ffi» + ii. In combining a number of different arcs of meridian, with the view of determining the figure of the earth, each aro will supply a number of equations in u and v and the corrections to its observed latitudes. Then, according to the method of least squares, those values of u and v are tho most probable which render the sum of the squares of all the errors x a minimum. The corrections x which are here applied arise not from errors of observation only. The mere uncertainty of a latitude, as determined with modem instruments, does not exceed a very small fraction of a second as far as errors of observation go, but no accuracy in observing will remove the error that may arise from local attraction. This, as we have seen, may amount to some seconds, so that the corrections x to the observed latitudes are attributable to local attraction. Archdeacon Pratt, in his treatise on the figure of the earth, objects to this mode of applying least squares first used by Bessel; but certainly Bessel was right, and the objection is groundless.

Comparisons of Standards. In determining the figure of the earth from the arcs of meridian measured in different countries, one source of uncertainty was, until the last few years, the want of comparisons between the standards of length in which the arcs were expressed. This has been removed by the very extensive series of comparisons recently made at Southampton (see Comparisons of Standard of Length oj England, France, Belgium, Prussia, Russia, India, and Australia, made at the Ordnance Survey Office, Southampton, 1866, and a paper in the Philosophical Transactions for 1873, by Lient-CoL A. R Clarke, C.B., R.E., on the further comparisons of the standards of Austria, Spain, the United States, Cape of Good Hope, and Russia). These direct comparisons, which were carried out with the highest attainable precision, are of very great value. The length of the toise has three independent determinations, viz., through the Russian standard double toise, the Prussian toise, and the Belgium toise,—giving for the length of tha toise, expressed in terms of the standard yard of England through the Russian standard 6 39453216 ft.

')„ „ Prussian „ 6-89458703 ft

- „ „ Belgian „ 6-39453215 ft

By combining all the different comparisons made in England and on the Continent on these bars, by the method of least squares, the final value of the toise is

6-39453348 ft (log = 0-8058088656), from which the greatest divergence of the three separate results specified above is only half a millionth of a toise, corresponding to ten feet in the earth's radius. From the known ratio of the toise and the metre, 864000 : 443296, we get for the metre

8-23086933 ft (log = 0-5150889356). That the close agreement between the determinations of the toise is not due to chance will be seen from the fact that the comparisons of the Prussian toise with the English standard involved 2340 micrometer readings and 520 thermometer readings, extending over twenty-five days, the probable error of the resulting length of the toise being * 0 00000015 yard. The probable error of the determination of the Belgian toise is * 0-00000027; that of the Bussian doable toise ± 0-00000031. With regard to tne metre, there is an independent determination resulting from the comparison of the platinum metreof the Royal Society,— a large number of observations giving for the length of the metre 3-28087206 feet, which differs from the former result by about one millionth part But this determination, involving the expansion of the bar for 30* of temperature, and being dependent on some old observations of Aragi, cannot be allowed any weight in modifying the result obtained through the toises. The Bussian standard, compared at Southampton, was that on which the length of their base lines and therefore their whole arc depends.

Calculation of tlte Semiaxet.

We now bring together the results of the various meridian arcs, omitting many short arcs which have been used in previous determinations, but which on account of their smallness have little influence in the result aimed at

The data of the French arc front Fonnentera to Dunkirk

Astronomical Latitudes.

rormcntera 38 89 53 17

Mountjouy 41 21 44-96

Barcelona 41 22 47-90

Carcassonne 43 12 64 80

Pantheon 48 60 47 98

Dunkirk 61 2 8-41

Distance of Parallel* Feet

982671 04 988701 '92 1657287-93 3710827-13 4509790 84

The latitude of Fonnentera as here given is taken from the observations of M. Biot, recorded and computed in the third volume of his Traiti Elimentaire d'Astronomie physique.

The latitude of the Pantheon, given in the Bast du Sijs&nu. Jlllrique Decimal (ii. 413), Is 48° 50'48"-86. In the Annates da I'Observotoire Imperial de Paris, vol. viii. page 817, ire find the latitude of south face of the observatory determined as 48° 50' H"71. The Pantheon being 35"-38 north of this, we thus get a second determination of its latitude. The mean is that given above.

The distance of the parallels of Dunkirk and Greenwich, deduced from the recent extension of the triangulatiun of England into France, in 1862, is 161407-3 feet, which is 3 9 feet greater than that obtained from Captain Rater's triangulatiou, and 3'2 feet les3 than the distance calculated by Delambre from General Boy's triangulation. The following table shows the data of the English arc with the distances in standard feet from Fonnentera.

_ . > s Feot


Greenwich 61 23 83-30 4671198-3

Arbury 62 18 26-59 4943837-6

Clifton 63 27 29-50 6894063-4

Kellie Law 56 14 63 00 64182217

Stirling 87 27 49 12 6857823-3

Saxavord 60 49 37-21 80868207

The latitude assigned in this table to Saxavord is not tlm directly observed latitude, which is 60° 49' 38"*58, for there are here a cluster of three points, whose latitudes are astronomically determined; and if we transfer, by means of the geodesic connection, the latitude of Gerik of Scate to Saxavord, we get 60° 49' 36"'59; and if we similarly transfer the latitude of Balta, we get 60° 49' 3fi"-46. The moan of these three is that entered in the above table.

For the Indian, arc in long. 77° 40' we have the following data:—

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And, finally, for the Peruvian arc, in long. 281° 0',

• Feet

Tarqui -3 4 32-068

Cotchesqui 0 2 31-387 1131036 3

Having now stated the data of the problem, we may eitbei seek that ellipsoid which best represents the observations, or we may restrict the figure to one of revolution. It will be convenient to commence with the supposition of an ellipsoidal figure, as on so doing we can, by a slight alteration in the equations of minimum, obtain also the required figure of revolution. It may be remarked that, whatever the icul figure may be, it is certain that if we presuppose it an ellipsoid, the arithmetical process will bring out an ellipsoid, which ellipsoid will agree better with all the observed latitudes than any spheroid would, therefore we do not prove that it is an ellipsoid; to prove this, arcs of longitude would be required. There is no doubt such arcs will bo shortly forthcoming, but as yet they are not available.

The first thing that occurs to one in considering an ellipsoidal earth is the question, What is a meridian curve I It may be defined in different ways: a point moving on the surface in the direction astronomically determined as "north" might be said to trace a meridian; or we may define it as the locus of those points which have a constant longitude, whose zeniths lie in a great circle of the heavens, having its poles in the equator; we adopt this definition. Let a, b, e be the scmiaxes, c being the polar semioxis. The equation of the ellipsoid being

if P be any point on tho surface, the direction cosines of the normal at P aro proportional to

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Hence the equation to a "parallel" in which the latitude rf> is constant is

So that in an ellipsoidal earth the parallel is no longer a plane curve. Let longitude be reckoned from the plane of xz. As there are two species of latitude, astronomical and geocentric, so there are in the ellipsoidal earth two species of longitude, geocentric (called u) and astronomical (called u). Conceive a line passing through the origin in the plane of thp equator and directed to a point whose longitude is Jir + w. The direction cosines of that line are—sin co, cos o>, and 0. Those points of the surfacs whose normals are at right angles to this line are in the meridian whose longitude is u; the condition of perpendicularity is expressed by

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and this, in fact, is the equation of the meridian, which is still on the ellipsoidal hypothesis a plane curve. The geocentric and astronomical longitudes are connected by the relation

•* tan «TMM tan m. This meridian curve is an ellipse whose minor semi.axis is e, and of which the semi-axis major is some quantity r intermediate between a and b, such that

l__cos* u sin* u

lake two quantities i.k, such that a'(l - i)-b\l + i) -iJ, then ** = r»(l -i cos 2«); and take » such that

r + e

and substitute the value of r, neglecting the square of i; this gives

n-t—- + 4 coa 2w .
k + c 4

Kow we have to determine not only the three semi-axes a, b, e, but the longitude of a. Let u, be the longitude of one of the measured meridian arcs, ufl the longitude of a, then, for that arc,



= J-j-^ +p cos 2«! + q sin 2ux,

where ip = i cos 2ti„ , 47 -1 si n 2 u,.

The normal at V does not pass through the axis of rotation, so that the observed latitudes on an ellipsoid are not exactly the quantities which should be used in the ordinary method of expressing the length of a meridian arc in terms cf the latitudes, lint it may be shown that this consideration may be neglected.

The data we have collected form 35 equations between the 40 ^-corrections to the observed latitudes, and the four unknown quantities determining the elements of the ellipsoid. Suppose nl to be an approximate value of the ratio k-c: t + c, so that


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e 3269 0
Longitude of a 16° 34' East.

The meridian of the greater axis passes, in the Eastern Hemisphere, through Spitzbergen, the Straits of Messina, Lake Chad in North Africa, and along the west coast of South Africa,—nearly corresponding to the meridian which passes over the greatest quantity of land in that hemisphere. In the Western Hemisphere it passes through Behring's Straits and through the centre of the Facile Ocean. The meridian (105* 34' E.) of the minor axis of the equator passes near North-east Cape on the Arctio Sea, through Tong-king and the Straits of Sunda, and corresponds nearly to the meridian which pa°ses over the greatest amount of land in Asia; and in the Western Hemisphere it passes through Smith Sound, ths west of Labrador, Montreal, between Cuba and Hayti, and along the west coast of South America, nearly coinciding with the meridian that passes over the greatest amount of land in that hemisphere.

'The length of the meridian quadrant passing through Paris, in the ellipsoidal figure given above, is 10001472-5 metres, showing that the length of the ideal French standard is considerably in error as representing the ten-millionth part of the quadrant The minimum quadrant, in longitude 105* 34', has a length of 10000024-5 metres. The probable error of the longitude of the major axis of the equator given above is of course large, as much perhaps as *16\

It has been objected to this figure of three unequal axes that it does not satisfy, in the proportions of the axes, the conditions brought out in Jacobi's theorem. Admitting this, it has to be noted, on the other hand, that Jacobi's theorem contemplates a homogeneous fluid, and this is certainly far from the actual condition of our globe, indeed the irregular distribution of continents and oceans suggests as possible a sensible divergence from a perfect surface of revolution.

If we limit the figure to being an ellipsoid of revolution, we get rid in our equations of two unknown quantities, on J the result may be expressed thus:—

Feet Uetrea

a = 20926062 = 6378206 4

c =20855121 = 6356503 8 e:a-293-98:294-98. • As might be expected, the sum of the squares of the 40 latitude corrections, viz., 153 99, is greater in this figure than in that of three axes, where it amounts to 138 30. In the Indian arc the largest corrections are at Dodagoontah, + 3"-87, and at Kalianpur, -3"68. In the Hussion nic the largest corrections are + 3"-76, at Tornea, and - 3'''31. at Staro Nekrassowka. Of the whole 40 corrections, 16 are under l"-0,10 between l"-0 and 2".0, 10 between 2" 0 and S"-0, and 4 over 3"'0. For the ellipsoidal figure the probable error of an observed latitude is * l"-42; for the spheroidal it would be very slightly larger. This quantity may be taken therefore as approximately the probable amount of local deflection.

In 1860, the Russian Government, at the instanco of M. Otto Struve, imperial astronomer at St Petersburg, invited the co-operation of the Governments of Prussia, Belgium, France, and England, to the important end of connecting their respective triangulatious so as to form a continuous chain under the parallel of 52* from the island of Valeutia on the south-west coast of Ireland, in longitude 10* 20' 40" W., to Orsk on the river Ural in Russia. This grand I undertaking was at ones sot in action, but up to the present

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