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.

Hyperbola
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.

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 ... ...