78 
ON THE SIX COORDINATES OF A LINE. 
[435 
Hence the line (a, b, c, f g, h) through the point p and in the plane P is a line 
the coordinates of which satisfy two linear relations as mentioned in the heading; and 
the theorem is thus proved. The demonstration would be simplified by taking, as is 
CL ry 
allowable, the homographic relation to be = k ^ . 
32. It appears from the foregoing examination of the case of two linear relations 
that in the following cases of three or more linear relations there is no real loss of 
generality in assuming that the coefficients of each set are the coordinates of a line; 
for if originally this be not so, we have only to replace the given relations by linear 
functions of these relations, and to assign such values to the multipliers X, X lt X 2 ... 
as in each case to make the new coefficients to be the coordinates of a line ; and as 
there are two or more arbitrary ratios X : X x : X 2 ... to be assigned at pleasure and 
only a single condition to be satisfied, no cases of failure can arise. The remaining 
cases may consequently be stated in a more simple form. 
Three linear relations, the coefficients of each set being the coordinates of a line. 
33. The three relations express that the line (a, b, c, f g, h) meets each of the 
three given lines ; that is, that the line is any generating line of a hyperboloid having 
the three given lines for directrices. 
Four linear relations, the coefficients of each set being the coordinates of a line. 
34. The four relations express that the line (a, b, c, f g, h) meets each of four 
given lines ; or what is the same thing, that the line is a tractor of four given 
lines. It is to be noticed that the four linear relations serve to express the ratios 
a : b : c : f : g : h linearly in terms of any one of these ratios, or what is the same 
thing, to express the several ratios in terms of an arbitrary ratio u : v. Substituting 
the resulting values in the equation 
af+ bg + ch = 0, 
we have a quadric equation for the determination of the remaining ratio, or of the 
ratio u : v ; and then each of the ratios of the coordinates can be expressed rationally 
in terms of either root of the quadric equation ; we thus obtain the coordinates of 
each of the two tractors of the four given lines ; or we have a complete analytical 
solution of the problem, to find the tractors of four given lines. The quadric equation 
may have equal roots ; that is, the four given lines may have a twofold tractor, which is 
then determined linearly. 
35. The theory of the linear relations of the coordinates (a, b, c, f, g, h) of a line 
may be considered in a different manner. It will be convenient to take the different 
cases in a reverse order, beginning with the extreme case (not before mentioned) of 
a fivefold relation and ascending to the case of a onefold or single relation.