In laying down Beacons or Buoys, their position is fixed, as may be seen in the Table in the Appendix, either by the intersection of two lines drawn through two leading objects on the shore (the magnetic bearings of which are given for the sake of easy reference on the spot, in finding out the marks), or by means of the angles contained between lines drawn to various objects on the shore, which meet at the Beacon or Buoy from which they are measured by means of a sextant. In the latter case, the angles are always measured around the whole horizon, thus affording a check by the difference of their sum from 360°. The magnetic bearing of one of those lines is afterwards carefully ascertained, by means of the prismatic compass (if possible from one of the objects on shore, and if not, conversely from the Beacon or Buoy), so as to afford the means of translating the whole into magnetic bearings for the use of seamen. The buoys are moored, as shewn in Plate XXXIII., by means of chains and iron sinkers, with a sufficient allowance in the length of the chain to permit them to _ride_ easily. APPENDIX. APPENDIX, No. I. TABLE OF CO-ORDINATES OF AN HYPERBOLIC COLUMN WHOSE DIAMETER AT THE TOP = 16 FEET, AT THE BASE = 42 FEET, AND HEIGHT = 120·25 FEET. The column is generated by the revolution of a rectangular hyperbola about one of its asymptotes. In the annexed figure (No. 98), _a_ _f_ is the height of the column, _a_ _c_ and _f_ _h_ the radii of its base and top; and we have to determine the particular hyperbola which will pass through the points _c_, _h_. Putting _b_ _e_ = _x_; _e_ _g_ = _y_, the equation to the curve, referred to its asymptotes, is _a_² _x_ _y_ = ----, 2 in which the value of the constant _a_² ---- 2 is to be found. For this purpose we have the conditions _a_ _c_ = 21; _f_ _h_ = 8; and _a_ _f_ = 120·25. Let the co-ordinates of the point _c_ be _x′_, _y′_, and of _h_, _x″_, _y″_, then _y′_ = 21; _y″_ = 8; _x″_ = _x′_ + 120·25. [Illustration: Fig. 98.] _a_² And since _x′_ _y′_ = ---- = _x″_ _y″_ 2 we have 21 _x′_ = 8 (_x′_ + 120·25) from which _x′_ = 74 _a_² and ---- = _x′_ _y′_ = 74 × 21 = 1554. 2 Therefore _x_ _y_ = 1554. Transferring the origin to _a_, _x_ becomes _x_ - _x′_ = _x_ - 74, and _y_ (_x_ - 74) = 1554, and the required equation by which the following Table was computed is, 1554 _y_ = --------. _x_ - 74 TABLE _of the Radii of the Hyperbolic Column at each foot of its Height_.