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dispose it in a manner simple and graceful, and calculated to dis- 
play the peculiar nature and construction of the edifice itself. 
The most common species of dome is a cupola, but it is far from 
being that which admits of the richest decoration, or which pro- 
duces the most pleasing effect. If the building be low, and the 
pase elliptic, a surbased ellipsoidal dome is probably the most ele- 
gant. If it be more elevated, it is necessary that the dome should 
be surmounted, because the encreased elevation will necessarily di- 
minish the apparent vertical height. The surfaces of domes are 
always however segments either of an ellipsoid, hyperboloid, or pa- 
raboloid. Of each of the latter surfaces there are two species, one 
only of which seems fitted for domes. I propose, in the following 
paper, to investigate the lines of curvature of these surfaces. ‘Those 
of the ellipsoid have already been determined by Monge ; however, 
as they can be all ultimately derived from the integration of the same 
equation, I shall include the ellipsoid to render the present investi- 
gation complete, 
Of the lines of curvature of surfaces of the second degree. 
The equation of those surfaces which have a centre related to the 
principal axes as axes of coordinates is 
az? + ay? + aa? 4+ Fr =0. 
To determine the differential coefficients of the first and second 
order, let this equation be differentiated considering = as a function 
of xy; the results will be 
