50 | 100 | 15° | 20° | 25° | 30° | Angles of inclination. 7.116 72.89 6.825 136.55 Exper. 1.-Centre of Gravity, 2.25 inches. 6.601 193.58 6.445 6.356 6-333 | Metacentre. 246-61 298.23 350.94 Momentum. Exper. 2.-Centre of Gravity, 3 inches. 61.66 114-17 | 160-22 | 202-51 | 243.75 | 286-48 | Momentum. Exper. 3.-Centre of Gravity, 4.50 inches. 39-19 69-40 | 93-48 | 114:33 | 134-79 | 157.57 | Momentum. Experiments with model 19. The sides incline inwards 45°. Length, 12 inches. Breadth, 9 inches. Total depth, 7 inches. Immersion, 4 inches. Weight of water displaced, 189-77 oz. 50 1. 100 5.544 5.240 40.13 72.57 | 15° | 20° | 25° | 300 | Angles of inclination. Exper. 1.-Centre of Gravity, 2.25 inches. 5.009 4.850 4.765 4.752 Metacentre. 99.80 124.31 | 149.34 | 174-85 Momentum. Exper. 2.-Centre of Gravitý, 3 inches. 30-99 | 54-36 | 72-66 | 88-45 | 101-24 | 122-43 | Momentum. Exper. 3.-Centre of Gravity, 4.50 inches. 12.72 17.96 | 18-40 | overset at 16° | Momentum. Metacentre by M. Bouguer's theorem, 6 inches. Experiments with triangular prism. (See model 20.) 12 inches. Breadth, 9 inches. Total depth, 12 inches. sion, 9 inches. Weight of water displaced, 279.53 oz. 50 I 10° 7.216 7.216 35.72 71.17 Length, 150 1 20° | 25° | 30° | Angles of inclination. Exper. 1.-Centre of Gravity, 5.75 inches. 29.63 59-03 | 87.99 | 116.28 | 143.68 | 169.98 | Momentum. Exper, 3.-Centre of Gravity, 6.25 inches. 23-54 | 46-90 | 69.90 | 92.37 114-14 | 135-04 | Momentum. Experiments with parabolic prism. (See model 21.) 12 inches. Breadth, 9 inches. Total depth, 7 inches. sion, 4 inches. Weight of water displaced, 186.35 oz. Length, 15° | 20° | 25° I 30° Angles of inclination. Exper. 1.-Centre of Gravity, 2.25 inches. 4.696 4.696 4.696 4.696 Metacentre, Exper. 2.-Centre of Gravity, 2.70 inches. 32.42 66.11 | 96-29 | 127.24 | 157.23 | 186-02 | Momentum. Exper. 3.-Centre of Gravity, 3 inches.. 27.55 | 54.89 | 81-82 | 108-12 | 133.60 | 158.06 | Momentum, Experiments with prism formed of circular segments. (See model 22.) Length, 12 inches. Breadth, 9 inches. Total depth, 7 inches. Immersion, 4 inches. Weight of water displaced, 178.22-oz, 5o 1 10° | 15° | 20° | 25° 130° | Angles of inclination. Exper. 1.-Centre of Gravity, 2.25 inches. 4.891 4.891 41.03 81.74 4.891 4.891 4.891 4.891 Metacentre. 121.84 161.00 リ 198.94 235.37 Momentum. Exper. 2.-Centre of Gravity, 2.87 inches. 31-40 | 62-56 | 93.24 | 123-21 | 152.24 | 180 12 | Momentum. Exper. 3.-Centre of Gravity, 3 inches. 29.38 | 58.5387 24 | 115.28 | 142.47 | 168.54 | Momentum. Experiments with compound figure. (See model and section 23.) Length, 12 inches. Breadth, 9 inches. Total depth, 71⁄2 inches. Immersion, 4 inches. Weight of water displaced, 199.02 oz. 50 I 10° | 15° 1 20° | 25° 1 30° | Angles of inclination. 4.873 49.39 4.873 98.41 Exper. 1.-Centre of Gravity, 2.025 inches. 4.873 .4.873 4.873 4.873 Metacentre. 146.68 193.83 239-51 283.37 Momentum. Exper. 2.-Centre of Gravity, 3 inches. 32-48 64-72 | 96-46 | 127-47157.50 | 186.34 | Momentum. Exper. 3.-Centre of Gravity, 4 inches. 15-1430-16 | 44.95 | 59.40 | 73.39 | 86.83 | Momentum. Exper. 4.-Centre of Gravity, 4.50 inches. 6:46 12.88 | 19.19 | 25-36 | 31-34 | 37.08 | Momentum. A, B, segment of a circle, of which E is the centre; B, C, being a tangent thereto; the angle, B, D, C, being 26° 34'. (See Plate XLV.) ARTICLE III. A Demonstration that the Ellipse, when viewed in a certain Position, appears circular. By S. SIP, (To Dr. Thomson.) Jan. 8, 1816. It is well known that the circle, if looked upon obliquely, will be projected into an ellipse; but I am not aware that the converse of this proposition has been demonstrated by showing that an ellipse, if viewed in a certain position, will appear circular. This has been established in the following theorems; but it was not the primary object with which they were drawn up. They were occasioned by the wish of getting a scalene cone turned truly in a lathe. Many good workmen assured me that it was impossible; and all the cones of this kind which I have seen (with the exception of that which I am about to mention) have been cut by hand. In thinking on the subject, it appeared that the best way would be to get a cone turned in the first instance with an elliptical base; but here again I met with a difficulty which was not surmounted till within these few months, by a very ingenious friend, who devised the means of executing exactly what I wished. By a mechanical method of trial, he afterwards found where he might cut this elliptical cone obliquely, so that the base should become a circle; but it seemed more satisfactory to investigate the problem mathematically, and I here send you the result. It would be unjust, however, not to add that the first lemma is taken from Dr. Robertson's Conic Sections (8vo. Oxford, 1802), and that several parts of the proposition were suggested by a recollection of the methods used in the first book of that valuable treatise. S. Lemma 1.-If from any point, I, of a straight line, NB (Plate XLV.* Fig. 1), a perpendicular, I L, be drawn; and if in all cases the rectangle under NI, I B, shall be equal to the square of I L, the curve passing through N L B shall be a semicircle. For bisect N B in C, and join C L. Then (5, ii. ) N I, IB, + ICC B2; therefore by hypothesis I L + I C2 = C B2; but (47, i.) I L2 + IC2 = C L2; consequently C B2 = C Lo. The same would hold wherever the point I is taken; therefore the locus of the points L must be a curve, which would be generated by the extremity, L, of the given straight line, CL, when its other extremity, C, is fixed, and the straight line revolves about it. Lemma 2.-Let VAB (Fig. 2) be an isosceles triangle, and O G a straight line longer than the base; then if VT be taken, such that O G2 A B2 : O G2 :: V B2: VT2, we shall have A B2: OG: AT, TB: VT. For A B : O G2 :: VT2 - V B2: VT; and when VC is drawn perpendicular to (and therefore bisecting) the base, V T V B = (47, i.) V C2 + C T2 — VC2 - CBCTC BAT, TB (6, ii.). , The lower division of the Plate, |