DE. PLUCKEE ON A NEW GEOMETET OF SPACE. 
781 
1 6. In denoting by y and v any two constant coefficients, 
Q=Q'+pQ''+»Q' w =0 (4) 
represents an infinite number ( oo 2 ) of complexes. All these complexes meet along the 
lines which simultaneously belong to any three of them, especially to 
O'=0, O"=0, O'"=0 (5) 
By means of these equations the position of such a line is determined, after having arbi- 
trarily assumed the value of one of the four constants upon which the line depends ; in 
other terms, three of these four constants are functions of the fourth, varying each by 
an infinitely small quantity if this one does. Hence we conclude that a line the coordi- 
nates of which verify the three equations (5), generates a surface in passing successively 
into all its positions. This surface (O', O", O'") is said to he represented hy the system of 
the three equations (5). 
17. Any point of space being given, there are three cones described by lines which 
belong to the three complexes (5) and pass through the given point. Generally the 
three cones (11) do not intersect along the same line. In certain positions only of 
the point they do. In this case their common intersection belongs to the surface 
(O', O", O'"), and therefore the point itself also. 
Put 
X' O' =F \_(x-x'\ (y—f), (z-z'), (yz'-y'z), -(xz'-x'z), (xy'-x'y)~\= 0, 
X"0" =F" [(x—x’), (y-y'), (z-z'), (, yz'-y'z ), -(xz'-x'z), (xy'-x'y)]=0, • 
X'"0" ' = F"' \_(x—x'), (y-y'), (z-z'), (yz'-y'z), -(xz'-x'z), (xy'-x'y)]=0. 
( 6 ) 
If x', y' , z' are referred to any arbitrary point, and x, y, z regarded as variable, these 
equations represent the three cones, (x’y'z') being their common centre, and their gene- 
rating lines belonging to the three complexes (5). Without changing the conditions of 
mutual intersection, the three cones may be moved parallel to themselves till the origin 
of coordinates becomes their common centre. After that displacement their equations 
are transformed into the following ones : 
F' \x, y, z, (yz'-y'z), -(xz'-x’z), (xy’-x'y)'] = 0, 
F" [x, y, z , (yz'-y'z), -(xz'-x’z), (xy’-x’y)]=0, (7) 
¥"[x, y, z, (yz'-y'z), -(xz’-x’z), (xy’-x’y)] = 0. j 
These equations being homogeneous with regard to (x, y, z), will, in the general case, 
not be simultaneously verified by the three variables. In order to express that they 
subsist simultaneously, we obtain, after having eliminated x, y, z, 
<p(x’, y', z')= 0, (8) 
<p indicating a function which involves the primitive constants of the three com- 
plexes (5). This function might be rendered homogeneous by introducing w'. This 
mdccclxv. 5 p 
