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
x y = a²2,
in which the value of the constant a² 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.
Fig. 98.
| And since | x′ y′ = a²2 = x″ y″ |
| we have | 21 x′ = 8 (x′ + 120·25) |
| from which | x′ = 74 |
| and | a²2 = x′ y′ = 74 × 21 = 1554. |
| 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, y = 1554 x - 74.
Table of the Radii of the Hyperbolic Column at each foot of its Height.