[916 
917] 
115 
917. 
58.] 
f the 
ullity 
[NOTE ON THE THEORY OF RATIONAL TRANSFORMATION.] 
[From the Proceedings of the London Mathematical Society, vol. xxn. (1891), 
pp. 475, 476.] 
In my paper, “ Note on the Theory of the Rational Transformation between 
Two Planes, and on Special Systems of Points,” Proc. Lond. Math. Soc. t. in. (1870), 
pp. 196—198, [450], I notice a difficulty which presents itself in the theory. The 
transformation is given by the equations 
x' : y' : z' = X : Y : Z, 
where X, F, Z are functions (*\x, y, z) n , such that X = 0, F=0, Z = 0 are curves 
in the first plane passing through a x points each once, a 2 points each twice (that is, 
having each of the ou, points for a double point), a 3 points each 3 times, and so 
on. We have as the condition of a single variable point of intersection, 
oq + 4ot 2 + 9a 3 + ... = ?г 2 - 1, 
and as the condition in order that each of the curves X = 0, F= 0, Z = 0, or say 
the curve aX + bY+ cZ = 0, may be unicursal, 
a 2 + 3« 3 + ... — £ (n — 1) (n — 2); 
and we thence deduce 
oq + 3a 2 + 6a 3 + ... = \n (n + 3) — 2; 
viz. the postulation of the fixed points quoad a curve of the order n is less by 2 
than the postulandum (or, as I prefer to call it, the capacity) ^n{n + 3) of the curve 
of the order n; that is, there are precisely the three asyzygetic curves X = 0, F= 0, 
Z= 0. This is as it should be, assuming that the (cq, a 2 , a 3 ,...) points are an ordinary 
system of points: but what if they form a special system having a postulation less 
15—2