DR. PLIJCKER ON A NEW GEOMETRY OF SPACE. 
787 
In order to fix the direction of a force, we may employ line-coordinates and choose the 
following, 
X, Y, Z, L, M, N, 
indicating the projections of the force on the three axes of coordinates OX, OY, OZ, 
and its three moments of rotation with regard to these axes. Between them the 
following equation of condition holds good, 
XL+YM+ZN=0 
(see No. 7). The quotients obtained by dividing any five of them by the sixth are the 
absolute values of coordinates. From these quotients the intensity of the force has dis- 
appeared. 
The same six constants , reduced by the last equation to five independent ones, may 
he regarded as the absolute values of the coordinates of the force. Instead of homoge- 
neous equations between them, if regarded as variable, representing complexes of lines 
(of directions of the forces), we now get ordinary equations between the same variables 
representing complexes of forces. 
The extension of all former developments thus indicated immediately occurs to us. 
A single instance may be referred to here. Forces constituting a linear complex are 
such passing in all directions through each point of space as have their intensity equal 
to the segments taken on their directions from the point to a certain plane corresponding 
to it. Forces common to two linear complexes and passing through a given point are 
confined within the same plane, the distance from the points where their directions meet 
a given line within the plane being then intensity. Forces, the coordinates of which 
verify simultaneously three linear equations, are distributed through space in such a 
manner that there is one force of a given intensity passing through each point of space, 
or, as we may add, confined in each plane. 
The general contents of this note (except § IV.) were in a verbal communication pre- 
sented by me at the last Birmingham Meeting of the British Association. As they 
concern the principles on which the original paper is based, giving to them a symmetry 
and a generality I was not before aware of, I thought it necessary to add the note 
to that paper. At the same time I also endeavoured to give an idea of the great ferti- 
lity of the method developed. But as I am now preparing a volume for publication on 
this subject, I do not think it suitable to enter here into any details. The work will 
embrace the theory of the general equation of the second degree between line-coordi- 
nates, requiring no means of discussion but those employed by me in the case of equa- 
tions of the same degree between point- or plane-coordinates. The complex of lines 
represented by such an equation may be regarded likewise as a complex of curves of the 
second class, one of which is confined in each plane, or as a complex of cones of the 
second order, each point of space being the centre of such a cone. In reducing the 
number of constants upon which the complex depends from 19 to 9, we pass in parti- 
