DE. PLtJCKEE ON A NEW GEOMETEY OE SPACE. 
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may likewise be regarded as line-coordinates. Here the equation of condition between 
the six coordinates becomes 
cos a cos X -J- cos j3 cos cos y cos v=0, 
which, added to the two following ones, 
cos 2 a+ cos 2 ]3-f- cos 2 y=l, 
cos 2 X cos 2 jU/+ cos 2 v = l, 
reduces to four the number of constants upon which the position of the line depends. 
6. The two sets of ratios I. and II. retain the same generality after putting w—w' — + 1, 
ot = gt'= + 1. If we suppose, again, that both planes and both points, by which the line is 
determined, are coincident, we get, choosing the under signs, two new sets of equal ratios, 
IV. —{udv—vdu) : —(tdv—vdt) : ( tdu—udt ) : dt : du : dv 
V. = dx : dy . dz : (ydz— zdy ) : — {xdz— zdx ) : {xdy— ydx). 
Thus we obtain two systems of differential coordinates, dx , dy , dz indicating the direction 
of the line, dt , du , dv the direction of the normal to the plane passing through it and 
the origin of coordinates. We may regard x, y, z, t, u, v as functions of time. 
7. We can represent the direction of a force by the right line, and its intensity by 
the distance of the two points by which the position of the line is fixed. In denominating 
the projections of the force on OX, OY, OZ by X, Y, Z, and the projections of its 
moment with regard to the origin on YZ, XZ, XY by L, M, N, we obtain the following 
new set of equal ratios : 
VI. =X : Y : Z : L : M : N. 
Therefore X, Y, Z, L, M, N may also be considered as six line-coordinates. The equa- 
tion of condition between them becomes 
XL+YM+ZN=0 (6) 
8. The six coordinates of each system range into two groups of three, to each 
coordinate of one group corresponds one of the other. By exchanging the three axes of 
coordinates, the three couples of corresponding coordinates are exchanged, both groups 
remaining the same. 
We may, in order to pass from the six coordinates of a right line to its five absolute 
coordinates, divide any five of them by the sixth. Here we meet two cases, in dividing 
either by a coordinate of the first or the second group. 
9. Let us divide the first two and the three last terms of the ratios I. by the third 
(tu 1 — t'u). In putting 
uv' — v!v tv 1 — t'v tu! — t'w uw' — u'w vw' — v'w 
tv!— t'u tu' — t'u tu' — t'u <7 ’ tu! —t'u tu! — t'u Yh 
where, according to the equation of condition (3), 
n=ra—sg, 
