DR. PLUCKER ON A NEW GEOMETRY OE SPACE. 
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62. In representing any three linear complexes by 
O =A r +Bs +C 4 -Do - +E§ +E (sg— rc)=0,j 
a , EEAV+B's+C , +D4+E'g+F(sg-rff)=0,i (89) 
0"=A"r+Bs" + C"+ D"<r + E"g + F "(sf -r<r) = 0, J 
the system of these three equations represents a linear configuration of rays. The com- 
plexes may be replaced by any three selected among those represented by 
O -J- /^O' -f- vO ! 1 = 0 
on giving to [h and v any values whatever. By combining the three complexes O, O', O" 
we get three congruencies, and accordingly three couples of directrices. Each ray of 
the configuration, belonging simultaneously to the three congruencies, meets both direc- 
trices of each couple. Hence in the general case the configuration is a hyperboloid ; its 
rays constitute one of its generations , while the directrices of all congruencies passing 
through it are right lines of its other generation. Any three directrices are sufficient in 
order to determine the hyperboloid. 
63. Let P and P', Q and Q', B and B' be the three couples of directrices, each couple 
determining a central plane. The three central planes II, K, P meet in one point C, 
which shall be called the centre of the configuration. The segment of any ray of a con- 
gruency bounded by both directrices being bisected by the central plane, the three right 
lines drawn through the centre C of the configuration to the three couples of directrices 
are bisected in the centre; they maybe called diameters of the configuration. 
Let, for instance, 7r and tt' be the extremities of that diameter, xCtt', which meets both 
directrices P and P'. The ray of the congruency (O, O') passing through 7 r is parallel 
to P', the ray passing through t 1 parallel to P. Both planes p andyV, drawn through P 
and P' parallel to the central plane n, each confining two right lines (one directrix and 
the ray parallel to the other) which belong to the two generations of the hyperboloid, 
touch that configuration, and the point where both right lines in each plane meet is the 
point of contact. 
Draw through the six directrices P and P', Q and Q', B and B' six planes p and p', 
g and q', r and n 3 parallel to the central planes H, K, P. The six planes thus obtained 
constitute a paralellopiped circumscribed to the configuration, the three diameters of 
which join each the points of contact within two opposite planes. The axes of the three 
corresponding congruencies (O, O'), (O, O'), (O', O") are equal to the distance of the 
three couples of opposite planes ; their centres are easily found. 
64. The hyperboloid thus obtained is not changed if the complexes O, O', O" be 
replaced by any three others taken among the complexes 
G+p,Q'+»>Q"=0, 
but the three congruencies vary, and their directrices and the three diameters of the 
hyperboloid. The directrices may be either real or imaginary ; accordingly the three 
mdccclxv. 5 M 
