DR. PLfiCKER OX A NEW GEOMETRY OF SPACE. 
755 
whence, in order to represent them in their primitive position, the following equations 
are derived, 
<7 — « + 0s — 0«r = 0, 
<r-\-a§—0s—0ar= 0 . 
By adding the two equations, after having multiplied the second by an undetermined 
coefficient p, the following equation results, 
(1 -\-[a)<7 — (1 — (1 — ^)0s— (1 -\-^)6ar— 0, 
which, on putting 
1=C=X. 
becomes 
a-Xag-\-X6s—6ar — 0 (81) 
By varying A all complexes intersecting each other along the congruency are repre- 
sented by this equation. Their axes are parallel to XY and meet OZ. According to 
(19) and (52) we may immediately derive the direction of the axes and their constants. 
The following way of proceeding leads us to the same results, giving besides the position 
in space of their axes. 
By turning OX and OY round OZ through an angle a, by means of the formula (34), 
in which a is to be replaced by a, the last equation is transformed into the following one, 
(cos u-\-Xa sin + (sin a — Xa cos co)£ + (X cos co -\-a sin u)6s' + (X sin u — a cos w)6r' = 0, 
whence, by putting 
we obtain 
tan co=Xa, 
(82) 
(1 + tan 2 co)a' -f- (X tan u — a)6r' + (X + a tan <y)0s' =0. 
Finally, by displacing the system of coordinates parallel to itself in such a way that the 
origin moves along OZ through z° , we get 
(1+ tan 2 !y)(7 , -|-(Xtan<y— a)6r'-\- (A+«tan^)0s'— (1-f- tan 2 <y)^V=0, 
whence, by putting 
there results 
„ A + a tan co , 
*= T+taAT* 
, A tan m — a . T . 
O '■ : ““ I i j. 2 * JC-T • 
I -f tan -to 
(83) 
(84) 
The values of tan <y, z°, and Ic remain real if both directrices become imaginary. In 
this case, XY always remaining the central plane of the congruency and OZ its axis, a, 
0, and ^ are to be replaced by a s/ — 1, $\/ —1, If a be real, we may put 
a= tan a, 
