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73. 
ABSTRACT OF A MEMOIR BY DR HESSE ON THE CONSTRUCTION 
OF THE SURFACE OF THE SECOND ORDER WHICH PASSES 
THROUGH NINE GIVEN POINTS. 
[From the Cambridge and Dublin Mathematical Journal, vol. iv. (1849), pp. 44—46.] 
The construction to be presently given of the surface of the second order which 
passes through nine given points, is taken from a memoir by Dr Hesse (Crelle, t. xxiv. 
[1842], p. 36). It depends upon the following lemma, which is there demonstrated. 
Lemma. The polar plane of a fixed point P with respect to any surface of the 
second order passing through seven given points, passes through a fixed point Q (which 
may be termed the harmonic pole of the point P with respect to the system of 
surfaces of the second order). 
Problem. Given the seven points 1, 2, 3, 4, 5, 6, 7, and a point P, to construct 
the harmonic pole Q of the point P with respect to the system of surfaces of the 
second order passing through the seven points. 
The required point Q may be considered as the intersection of the polar planes 
of the point P with respect to any three hyperboloids, each of which passes through 
the seven given points; any such hyperboloid may be considered as determined by 
means of three of its generating lines. These considerations lead to the construction 
following. 
1. Connecting the points 1 and 2, and also the points 3 and 4, by two straight 
lines, and determining the three lines, each of which passes through one of the points 
5, 6, 7, and intersects both of the first-mentioned lines, the three lines so determined 
are generating lines of a hyperboloid passing through the seven points. 
C. 
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