GOG E A II T II through the Russian standard ........ 6 39453216 ft. ,, Prussian ,, ......... 6 39453703 ft. ,, ,, Belgian ,, ......... 6 39453215 i t. 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 = 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 3-28086933 ft. (log = 5159889356). 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 deter mination of the Belgian toise is 0-00000027; that of the Russian double toise 00000031. With regard to the metre, there is an independent determination resulting from the comparison of the platinum metre of 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 Arago, cannot be allowed any weight in modifying the result obtained through the toises. The Russian standard, com pared at Southampton, was that on which the length of their base lines and therefore their whole arc depends. Calculation of the Semiaxes. 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 from Formentera to Dunkirk are Stations. Astronomical Latitudes. Formentera 38 39 5317 Mountjouy 41 21 44 96 Barcelona 41 22 47 90 Carcassonne 43 12 54 30 Pantheon 48 50 47 98 Dunkirk... . 51 2 8 41 Distance of Parallels. Feet. 982671-04 988701-92 1657287-93 3710827-13 4509790-84 The latitude of Formentera as here given is taken from the observations of 11. Biot, recorded and computed in the third volume of his TraiU Elementaire d Astronomic physique. The latitude of the Pantheon, given in the Base, du Sysiemt, Melrique Decimal (ii. 413), is 48 50 48"-86. In the Annalcs de 1 Obscrvatoire Imperial de Paris, vol. viii. page 317, we find the latitude of south face of the observatory determined as 48 50 11" 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 Kater s triangulation, and 3 2 feet less than the distance calculated by Delambre from General Roy s triangulation. The following table shows the data of the English arc with the distances in standard feet from Formentera. . / a Feet. Formentera ............. Greenwich .............. 51 28 38 30 4671198 3 Arbury .................. 52 13 26 59 4943837 6 Clifton .................. 53 27 29-50 5394063 4 Kellie Law ............. 56 14 53 GO 6413221-7 Stirling ................. 57 27 49 12 6857323 3 Saxavord ............... 60 49 37 21 8086820-7 The latitude assigned in this table to Saxavord is not the 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 Gerth of Scaiv to Saxavord, we get 60 49 36" 59; and if we similarly transfer the latitude of JMta, we get 60 49 36"-46. The mean of these three is that entered in the above table. For the Indian arc in long. 77 40 we have the follow ing data : Feet. Punnce 8 9 31132 Futehapolliam 10 59 42 276 1029174 9 Dodagoontah 12 59 52 165 1756562 Namthabad 15 5 53 562 2518376 3 Daumergida 18 3 15 292 35S1788 4 Takalldiera 21 5 51 532 46 ( J7329 5 Kalumpur 24 7 11 262 57946957 Kaliaua 29 30 48 322 7755S35 9 The data of the Russian arc (long. 26 40 ) taken from M. Struve s work are as below : / // Feet. Staro Nekrassowka.. 45 20 2 94 "Wodolui 47 1 24-98 616529 81 Ssuprunkowzi 48 45 3 04 1246762-17 Kremeiietz 50 5 49 95 1737551 48 Belin 52 2 42 16 2448745 17 Nemesch 54 39 4 16 3400312 63 Jacobstadt 56 30 4 97 4-076412-28 Dorpat 58 22 47 56 4762421-43 Hogland 60 5 9 84 5386135 39 Kilpi-maki 62 38 5 25 6317905 67 Tornea 65 49 44 57 7486789 97 Stuor-oivi 68 40 58 40 8530517 90 Fuglences 70 40 11 -23 9257921-06 From the arc measured by Sir Thomas Maclear in long. 18 30 , we have / Feet. North End 29 44 17 66 Ileerenlogement Berg. 31 58 9 11 811507 7 Royal Observatory.... 33 56 3 20 1526386 8 ZwartKop 34 13 3213 16325S3 3 Cape Point 34 21 6 26 167S375 7 And, finally, for the Peruvian arc, in long. 281 , Feet. Tarqui -3 4 32-068 Cotchesqui 2 31 387 1131036 3 Having now stated the data of the problem, we may either seek that ellipsoid which best represents the observations, or we may restrict the figure to one of revolution. It will be con venient 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 real 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 be 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 1 It may be defined in different ways : a point moving on tLe 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, 6, c be the semiaxes, c being the polar semiaxis. The equation of the ellipsoid being ?L + VI + z i f if P be any point on the surface, the direction cosines of
the normal at P are proportional to