DR. PLUCKER ON A NEW GEOMETRY OE SPACE. 
783 
tical expression of these new conditions, we may, as we did before, replace both cones 
by such as have the origin as centre. In putting 
the equations of these new cones may be written thus (No. 17), 
f(?> <b z')=°,\ 
/£p, ( b y\ 
(ii) 
/"and/ 7 representing two functions of the variables^) and q, by means of which the lines 
constituting the two cones are determined, x 1 , y\ z! being the coordinates of the given 
point. If two of the mn intersecting lines of the two cones are coincident along any right 
line (p, q), we get for the determination of that line, besides the two equations (11), the 
following new one, 
£ 
dq 
which, if expanded, likewise assumes the form 
f"(p, q, x’, y\ *>=0, 
( 12 ) 
f n indicating a new function. By eliminating p and q between the three equations 
(11) and (12), we get an equation of the form 
W,y',z')= 0, (13) 
representing, if at, y\ z' be regarded as variable, a developable surface , the locus of those 
points through which double lines of the congruency pass, or, in other terms, the locus 
of the double lines themselves. 
In supposing that three intersecting lines of the two cones (11) fall within the same 
line (p, q), the following new equation of condition is obtained 
dj_ df 
dp* \dq) dpdq dq dp dq 9 - \dp) dp dq 
dtf(dfY_ o d?f df df dj 1 / df'\ > = df-7f 
dp* \dq) dpdq dq' dp ' dq 2 \dp) dp dq 
which again may be expanded into an equation of the form 
f"(p, q,x',y', z ')= 0 (14) 
This equation, combined with the three former equations (11) and (12), furnishes a new 
equation of condition, 
t(x',y',z ')= 0 (15) 
The system of the two equations (13) and (15) gives, as locus of points through which 
triple lines of the congruency pass, a curve of double curvature. 
In pursuing the same course a new equation of the same form as (13) and (15) is 
5 p 2 
