'220 
ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. 
[447 
correspond to G", H", I", and between the first and second figures make D, E, F 
correspond to D', E', F', and G, H, I to G', H', I', the diagram being thus: 
First Fig. 
Second Fig. 
Third Fig. 
Fourth Fig. 
A, B, G 
A', B', G' | 
A", B", G" 
A'", B'", G'" 
D, E, F 
D', E’, F' 
D", E", F" 
D"’, E"', F" 
G, H, I 
G', H', 1' 
G", H", I" 
G'", H'", I"' 
Observe that in the principal systems (for instance, A, B, G and A', B', O') the points 
A, B, G correspond, not to the points A', B', C, but to the lines B'G', G'A', A'B' 
respectively; and so in the other case. 
64. Consider now a line in the first figure: there corresponds hereto in the 
second figure a conic through the points A', B', G'; and to this conic there corre 
sponds in the third figure a quartic curve passing through each of the points 
A", B", G" once, and through each of the points D", E", F" twice. And conversely, 
to a line in the third figure corresponds in the second figure a conic through the 
points D', E, F'; and hereto in the first figure a quartic through the points D, E, F, 
once and through the points A, B, G twice; that is, we have between the first and 
third figures a quartic transformation wherein a 4 = cl, = 3 and a/ = a.! = 3, or say a 
quartic transformation 3^2 and 3 r 3 2 . In like manner, passing to the fourth figure, to 
a line in the first figure corresponds in the fourth figure an octic curve passing 
through A"’, B'", G" once, through I)'", E'", F"' twice, and through G"', H'", I"' four 
times; and conversely, to a line in the fourth figure there corresponds in the first 
figure an octic curve passing through the points G, H, I once, the points D, E, F 
twice, and the points A, B, G four times; that is, between the first and fourth figures 
we have an octic transformation, wherein a 1 = a. 2 = a 4 = 3, a/ = a./ = a/ = 3, or say a 
transformation, order 8, of the form 3 1 3 2 3 4 and SfiVk. And so between the first and 
fifth figures there is a transformation, order 16, of the form dJbSjlg and 3 1 3 2 3 4 3 8 . 
65. It is, moreover, easy to find the Jacobians or counter-systems in the several 
transformations respectively. Thus, in the transformation between the first and second 
figures, in the second figure the Jacobian consists of 3 lines such as B'G' (viz., these 
are, of course, the lines B'G', G'A', A'B'). Hence, in the transformation between the 
first and third figures, the Jacobian in the third figure consists of 
3 conics B"G"(D"E"F"), 
3 lines D"E"; 
viz., one of the conics is that through the five points B", G", D", E", F", one of the 
lines that through the two points T>", E". And so in the fourth figure, the Jacobian 
consists of 
3 quartics B'"G"'(D'"E"'F"') 1 {G , "H'"I"% 
3 conics 
3 lines 
D'"E' 
(G"'H"T") 1 , 
G"'H"' : 
lines