756 
DE. PLtj CKEE ON A NEW OEOMETEY OF SPACE. 
2a being the angle between the directions of the two directrices, bisected by XZ. 
Accordingly we get 
y tan co 
tan «’ 
(85) 
>0 1 + tan 2 « tan co 
tan « 1 + tan 2 co 
„ sin co cos co 
= 0- 
sin « cos sc 
. sin 2co 
= 4 • 0 , 
sin ‘Jet ’ 
Jc—6 
tan 2 a — tan 2 co 
tan «(1 + tan 2 co) 
] sirr a cos^ co — sin‘ co cos^a 
sin a cos a 
^sin (a + co) sin (a — co) 
sin a cos a 
( 86 ) 
(87) 
The expression of z° shows that the axis within the central plane is directed along 
one of the two right lines bisecting, within this plane, the angle between the directions 
of the two directrices. These two right lines, having a peculiar relation to the congru- 
ency, may be called its second and third axis. The three axes, perpendicular to each 
other, meet in the centre of the congruency. 
In order to express the angle a by means of 2 °, we get the following equation, 
2° 
sin 2cy= - sin 2a, 
0 
indicating two directions perpendicular to each other, and corresponding to any value 
of 2°. 
61. By replacing in the expression 
0 tanw 
sin « cos a 1 + tan 2 w 
tan a by v - , we obtain on omitting the accent of 2 °, 
z(f+x*)= 
sin « cos cl 
xy. 
( 88 ) 
The axes of all complexes constituting the congruency are confined within the surface 
represented by that equation. But this equation remaining unaltered if the axes OX 
and OY are replaced by one another, it is evident that the same surface contained the 
axes of two different series of complexes ; one of the two series constituting the given 
congruency, while the other constitutes a strange one, obtained by turning the given 
congruency round its axis through a right angle. 
