214 
ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. [447 
directly: taking the coordinates (x, y, z) to refer to the principal system of the first 
plane (viz., taking the three points to be the vertices of the triangle formed by the 
lines x = 0, y= 0, z = 0), then X = 0, F=0, Z=0 being conics through the three points, 
the functions X, Y, Z will be each of them of the form fyz + gzx + lixy; x, y', z 
being proportional to three such functions, there will be linear functions of x', y', z 
proportional to yz, zx, xy; or taking these linear functions of the original (x, y', z') for 
the coordinates (xf, y, z') of a point in the second plane, the formulae of transformation 
will be x’ : y' : Y — yz : zx : xy, and we have then conversely x : y : z = y'z' : z'x' : xy ; 
that is, the formulae for the transformation in question are 
x : y : / = yz : zx : xy, and x : y : z — y'z' : z’x : xy. 
We at once verify a posteriori that the Jacobian in the first plane is xyz = 0, and 
that in the second plane is xy'z' = 0. 
The equations may be written 
/ / / 
x : y : z 
- : - : -. and x : y 
X y z 
1 
' y' 
1 
or, if we please, xx' = yy' — zz'; the transformation is thus given as an inverse trans 
formation. 
{49. With respect to the metrical interpretation and actual construction of the 
transformation, it is to be observed that if x, y, z be taken to be proportional (not 
to given multiples of the perpendicular distances, but) to the perpendicular distances 
of P from the sides of the triangle in the first plane, and similarly x', y', z' to be 
proportional to the perpendicular distances of P' from the sides of the triangle in the 
second plane, then in general the equations of transformation must be written, not as 
above, but in the form X °l =—= ^-, involving arbitrary multipliers f : g : h. We may 
imagine in the second plane a point P" determined by coordinates (x", y", z"),—the 
same coordinates as {x, y', z'), that is, proportional to the perpendicular distances of 
P" from the sides of the triangle in the second plane,—which point P" corresponds 
homographically to P in such wise that j. : ^ ^ =x" : y" : z". We have then, in 
the second plane, the two points P', P" corresponding to each other in such wise 
that x'x" = y'y” = zz"; and either of these points being given, the other can at once 
be constructed; viz., it is obvious that, joining P', P" with any vertex, say A', of the 
triangle A'B'C', the lines A'P', A'P" are equally inclined to the bisectors of the 
angle A'; and consequently, P' being given, we have the three lines A'P", B'P", CP" 
intersecting in a common point P", which is therefore determined by means of any 
two of these lines. We have thus a geometrical construction of the transformation 
between P and P'.} 
50. The analysis assumes that the principal points A, B, C of the first figure 
are three distinct points; but they may two of them, or all three, coincide. In the 
first case, say if B, C coincide, the line BC is still to be regarded as having a 
definite direction; and taking x = 0 for this line, y = 0 for the line joining A with