786 
DE. PLUCKER ON A NEW GEOMETRY OE SPACE. 
where we may suppose that P, Q, E, S are the former four axes of coordinates ; x, x!, x, X', 
I&&, v , v' indicate any constant coefficients. 
In putting the coordinates q, r, s, t, u . . equal to zero, the general equations of the 
complexes become 
These new equations represent complexes of a peculiar kind, the lines of which inter- 
sect their axes ; they may be said to represent the axes themselves. 
In order to satisfy the equation (18), we put 
whence 
H p 4- X'H ? + gJ a r + v' a s , J 
(19) 
t —xj) -f-X^ -f -[hr +w, 1 
u=xp+'k'q+[A'r-\-v's. j 
( 20 ) 
The equations (19) require that the origin met by the axes P, Q, E, S be likewise met 
by the new axes T, U . . . 
Therefore q, r, s, t, u . . may be regarded as coordinates of the right line along 
which all complexes meet ; the axes of the complexes intersecting the same right line 
being the axes of coordinates. A right line being completely determined by the first 
four coordinates, those remaining depend upon them by linear equations (20). 
The system of four axes of coordinates depends upon 16, of five axes upon 19, of six 
upon 22 constants. 
Having thus established a system of coordinates which, independently of points and 
planes, fixes the position of a right line in space, we are enabled, by regarding right lines 
as elements of space, to reconstruct the whole geometry without recurring to the ordi- 
nary system. Here we are guided by analogy. As far as I may judge, the task is a 
most grateful but at the same time a long and laborious one. 
IV. Geometry of Forces. 
25. In recapitulating the contents of the first three paragraphs of this note, new con- 
siderations have been suggested to me, which seem calculated, while greatly increasing 
again this kind of inquiry, to put the key-stone to it. Hitherto, when I borrowed 
technical terms from mechanical science, the only intention was to simplify the expression. 
But force may be regarded as a merely geometrical notion, and there is only one step 
more to be taken in order to arrive at a “ Geometry of Forces ,” as there is a geometry 
based on the notion of right lines. 
Forces depend upon five independent constants, four of which indicate their position, 
while the fifth indicates their intensity. We may call these constants the five coordi- 
nates of the forces. 
