219 
ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. 
points P, P' will therefore contribute equal values to the deficiencies of the two curves 
respectively ; so that, in equating the two deficiencies, we may disregard P, P', and 
attend only to the points a 1 , a 2 ,...oin-i of the first plane, and a/, a 2 ', ...aV-i of the 
second plane. The required relation thus is 
i (k - 1) (k - 2) - 2 J jet, (a, - 1) + a, (a s - 1)... +<*„_, (a„_, - 1)) 
= i (V - 1 ) W - 2) - 2 \ K « - 1) + ai (a: - 1)... + a'«., (a'_, - 1 )(. 
61. In the case of the quadric transformation n = 2, we have in the first plane 
the three points a 1} say these are A, B, G; and in the second plane the three points 
a/, say these are A', B', O'. And if in the first plane the curve of the order k 
passes a, b, c times through the three points respectively, and in the second plane 
the corresponding curve of the order k' passes a, b', c times through the three points 
respectively, then it is easy to obtain 
k! = 2k — a — b — c, 
a = k'-b'-c', 
b — k' — d — a', 
c = k' -a - V. 
a' = k — b — c, 
b' = k — c — a, 
c = k — a — b. 
The Quadric Transformation any number of times repeated. 
62. We may successively repeat the quadric transformation according to the type: 
Third Fig. 
Second Fig. 
A', B', C 
D', E\ F 
Fourth Fig. 
First Fig. 
A, B, C 
viz., in the transformation between the first and second figures, the principal systems 
are ABC and A'B'C' respectively; in that between the second and third figures, 
they are D'E'F' and D"E"F" respectively; in that between the third and fourth 
figures, they are G"H"I" and G'"H"'T"; and so on. And it is then easy to see that 
between the first and any subsequent figure we have a rational transformation of 
the order 2 for the second figure, 4 for the third figure, 8 for the fourth figure, and 
so on. 
63. But to further explain the relation, we may complete the diagram, by taking, 
in the transformation between the second and third figures, A", B", G" to correspond 
to A', B', C'; similarly, in that between the third and fourth, A'", B'", G'" to 
correspond to A", B", G" ; and D"', E”', F"' to correspond to D", E", F". And so in 
the transformation between the second and third figure, we may make G', H', F 
28—2