DR. PLUCKER ON A NEW GEOMETRY OE SPACE. 
789 
complex may be regarded as two congruent conjugate lines. Principle of polar reci- 
procity applied, 20. Construction of the plane corresponding to a given point, of the 
point corresponding to a given plane, 21, 22. Geometrical determination of the con- 
stant of the general equation of the complex. There is a characteristic direction given 
by the double equation ^-= = If that direction falls within xy , the term (sg—rv) 
disappears and the general equation becomes linear. If any plane perpendicular to it is 
taken as one of the three planes of rectangular coordinates, and the corresponding point 
within it as origin, the general equation assumes one of the forms, s=kg, r=ka, 
sg—rtr=k. A linear complex may, without being altered, turn round a fixed line, and move 
along it parallel to itself, 23-29. Geometrical interpretation of the last equations, 30. 
Points and planes corresponding to one another with regard to the complex sg—ra=.k. 
Geometrical interpretations, 31, 32. Generalization, 33. Conjugate lines with regard 
to the complex sg — r<r=k, 34. A linear complex depends upon five constants, four of 
which give the position of its axis, 35. Formulse of the transformation of ray-coordinates 
corresponding to any displacement of the axes of coordinates, 36-38. Analytical deter- 
mination of the axis of a complex, represented by the general equation. Determination 
of k, 39-43. In the peculiar case in which k is equal to zero, all rays meet the axis of 
the complex, 44. Rays passing through the same point, 45. 
Linear congruencies of rays. — A linear congruency, along which an infinite number of 
complexes intersect each other, is represented^ by the equations of any two of them. 
Through a given point of space only one ray passes, corresponding to that point, as there 
is only one ray confined within a given plane, 46. There is in each complex passing 
through the congruency one line conjugate to a given right line : all these lines belong 
to one generation of a hyperboloid, the second generation of which contains rays of the 
congruency. Generation of a linear congruency by a variable hyperboloid, 47-49. 
Characteristic section of a congruency to which the axes of all passing complexes are 
parallel. The axis of the congruency is a fixed right line, perpendicular to that section 
on which the axes of all complexes meet at right angles, 50, 51. The locus of points 
having in all complexes the same corresponding plane is a system of two right lines, the 
directrices of the congruency. Central plane parallel to both directrices and equidistant 
from them. The directrices may be real or imaginary, 52-54. In the first case there 
are amongst the complexes two of a peculiar description [44] having both directrices as 
axes. All rays of the congruency meet both its directrices, 55. The peculiar case in 
which one of the two directrices is infinitely distant, 56. Each of any two complexes 
being given by means of its constant k and the position of its axis, to determine both 
directrices of the congruency, 57-59. A congruency being given by means of its two direc- 
trices, to determine the constants and the axes of the complexes passing through it. 
Centre of the congruency. The two secondary axes within the central plane, 60. Locus 
of the axes of all complexes meeting along the same congruency, 61. 
Linear configurations of rays represented by the equations of three linear complexes. 
mdccclxv. 5 Q 
