Chapter XXV: 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.

_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_.

+-------+-------++-------+-------++-------+-------++-------+-------+
|Height.|Radius.||Height.|Radius.||Height.|Radius.||Height.|Radius.|
+-------+-------++-------+-------++-------+-------++-------+-------+
| 0 | 21·000|| 31 | 14·800|| 62 | 11·426|| 93 | 9·305 |
| 1 | 20·720|| 32 | 14·660|| 63 | 11·343|| 94 | 9·250 |
| 2 | 20·447|| 33 | 14·523|| 64 | 11·261|| 95 | 9·195 |
| 3 | 20·182|| 34 | 14·389|| 65 | 11·180|| 96 | 9·141 |
| 4 | 19·923|| 35 | 14·257|| 66 | 11·100|| 97 | 9·088 |
| 5 | 19·671|| 36 | 14·127|| 67 | 11·021|| 98 | 9·035 |
| 6 | 19·425|| 37 | 14·000|| 68 | 10·944|| 99 | 8·983 |
| 7 | 19·185|| 38 | 13·875|| 69 | 10·867|| 100 | 8·931 |
| 8 | 18·951|| 39 | 13·752|| 70 | 10·792|| 101 | 8·880 |
| 9 | 18·723|| 40 | 13·632|| 71 | 10·717|| 102 | 8·830 |
| 10 | 18·500|| 41 | 13·513|| 72 | 10·644|| 103 | 8·780 |
| 11 | 18·282|| 42 | 13·397|| 73 | 10·571|| 104 | 8·730 |
| 12 | 18·070|| 43 | 13·282|| 74 | 10·500|| 105 | 8·681 |
| 13 | 17·862|| 44 | 13·170|| 75 | 10·430|| 106 | 8·633 |
| 14 | 17·659|| 45 | 13·059|| 76 | 10·360|| 107 | 8·586 |
| 15 | 17·461|| 46 | 12·950|| 77 | 10·291|| 108 | 8·539 |
| 16 | 17·267|| 47 | 12·843|| 78 | 10·224|| 109 | 8·492 |
| 17 | 17·077|| 48 | 12·738|| 79 | 10·157|| 110 | 8·446 |
| 18 | 16·891|| 49 | 12·634|| 80 | 10·091|| 111 | 8·400 |
| 19 | 16·710|| 50 | 12·532|| 81 | 10·026|| 112 | 8·355 |
| 20 | 16·532|| 51 | 12·432|| 82 | 9·962|| 113 | 8·310 |
| 21 | 16·358|| 52 | 12·333|| 83 | 9·898|| 114 | 8·266 |
| 22 | 16·188|| 53 | 12·236|| 84 | 9·835|| 115 | 8·222 |
| 23 | 16·021|| 54 | 12·141|| 85 | 9·774|| 116 | 8·179 |
| 24 | 15·857|| 55 | 12·046|| 86 | 9·712|| 117 | 8·136 |
| 25 | 15·697|| 56 | 11·954|| 87 | 9·652|| 118 | 8·094 |
| 26 | 15·540|| 57 | 11·862|| 88 | 9·593|| 119 | 8·052 |
| 27 | 15·386|| 58 | 11·773|| 89 | 9·534|| 120 | 8·010 |
| 28 | 15·235|| 59 | 11·684|| 90 | 9·476|| 120·25| 8·000 |
| 29 | 15·087|| 60 | 11·597|| 91 | 9·418|| ... | ... |
| 30 | 14·942|| 61 | 11·511|| 92 | 9·361|| ... | ... |
+-------+-------++-------+-------++-------+-------++-------+-------+