754 
DR. PLtjCKER ON A. NEW GEOMETRY OE SPACE. 
On denoting the roots of these equations by z' sinS, z " sin $•, and (^j , (^j , we obtain 
(k—k 1 ) cos $ + A sin 3- 
sin S' 
4 M'+ [(A: — k 1 ) cosS — A sin S] ? 
sin 2 S 
{z'-z'J 
(y\ ( y\" (A‘ + #)cos5 — AsinS 
a y\' fy\"\ 2 4^'+ [(A — A') e°s ^ — A sin •&] 
*) " w ) ~ 
The roots of both equations are simultaneously either real, or imaginary, or congruent. 
In the last case we have 
(k—k') cosS — A sin$-=2v^ — kk/, 
whence 
(f)'-(5W4- 
The central plane of the congruency is represented by 
( k — k') cos S — A sin S 
2 sin S 
(SO) 
In two peculiar cases this equation becomes 
z —\ A, 
either if 
&=**■, 
or, whatever may be if 
k=U. 
Hence the axes of any two complexes selected among those intersecting each other 
along a given congruency are at equal distances from its central plane if their directions 
are perpendicular to each other, or if the constants of both complexes are the same. 
60. Without entering into a more detailed discussion of the last results we may 
finally treat the inverse problem : a congruency being given by means of its two direc- 
trices, to determine the complexes passing through it. On the supposition of rectangular 
coordinates, the two directrices may be represented by the following systems of equations, 
y—ax— 0, z=d,. 
y-\-a%=0, z=—0. 
These directrices are the axes of two complexes of a peculiar description, ranging among 
the infinite number of complexes which intersect each other along the congruency. 
The two complexes, if moved parallel to themselves till their axes fall within XY, are 
represented by the equations 
<7 — ag= 0, 
<T -|-ff£ = 0, 
