Note 12 — Chapter III · ON THE CONSTRUCTION OF LIGHTHOUSE TOWERS.

The solid, in this case, would have been formed by the revolution of the interior conchoid of Nicomedes about its directrix; and its co-ordinates were kindly calculated for me by my late revered preceptor, Dr Wallace, Professor of Mathematics in the University of Edinburgh, who employed so many hours of his latter years in labours of kindness among his friends. This act of the Professor was the result of a conversation I had with him on the subject. Before I received his friendly communication, however, I had resolved to adopt the rectangular hyperbola, whose co-ordinates I had myself determined with this view some time before; and when I found that the conchoid and the hyperbola, traced between the two fixed points by means of the calculated co-ordinates, were so nearly coincident, that it was difficult to prevent their running into each other, even when drawn out on a large scale, I determined to adhere to my original purpose of adopting the latter curve as my guide.

[13] The last column of this table is derived as follows:—Assuming that the economic advantage of any proposed tower of given height and diameter at base and top, is inversely as the mass and the height of the centre of gravity above the base, and denoting these quantities by M and G respectively, the fraction 1G·M may be taken as an indication of the economic advantage of the proposed tower. Let 1G′·M′ express the economic advantage of another tower; then the advantage of the second tower, compared to that of the first, taken as unity, will be G·MG′·M′, by which expression, the last column in the table was calculated.