254 
NOTE ON THE THEORY OF THE RATIONAL TRANSFORMATION [450 
I believe the better course is to assume (1) and (3) as the fundamental equations, 
from them deducing (2); and we thus also get over a difficulty presently referred to, 
but which did not occur to me when the. memoir was written. 
In fact, starting with the equations x' \ y' \ z' — X : Y : Z (which are to give 
x : y : z — X' : Y' : Z'), we have in the first instance the equation (1). Moreover, 
establishing for x', y', z' a linear equation ax' + by' + cz' = 0, we have corresponding hereto 
a curve aX + bY+cZ = 0, and the coordinates x, y, z of a point on this curve are 
proportional to X' : Y' : Z'; that is, substituting for z the value — - {ax' + by), they 
c 
are proportional to rational and integral (homogeneous) functions of (x\ y'), that is, to 
rational and integral functions of the single parameter x' : y'; wherefore the curve 
aX + bY + cZ = 0 is unicursal; whence the equation (3). The like change may be 
made in the theory of the rational transformation between two spaces; and it is in 
this case a more important one. 
The difficulty is as follows: It is not self-evident that we are at liberty to assume 
Qj d~ 3a 2 d- Bcty,.. ^ -2 (?l 2 d" 3n) — 2 J 
for imagine that we had a system of (aj,' a 2 , a 3> ...) points, such that a 2 d- 4a. 3 d- • ■ • being 
= n 1 — 1, and a 2 d- 3a 2 d- • • • being > ^ {n 2 d- Sri) — 2, the points were such that the conditions 
in question (viz., the condition that the curve passes once through each of the points a 2 , 
twice through each of the points ot 2 ...) should be less than a 2 + 3a 2 d- •••, and in fact 
= or < | ('n 2 d- Sri) — 2 ; then the functions X, Y, Z would not of necessity be connected 
by a linear relation \X +/¿Y + vZ = 0, and the ground for the assumption in question, 
a 2 -+• 3a 2 + ...^\{ri d- Sri) — 2, would no longer exist. And except by the process now 
adopted of deriving the equation (2) from the equations (1) and (3), I do not know 
how the impossibility of such a system is to be established; viz., I do not know how 
we are to prove the following theorem:—There is not any system of (a 2 , a 2 , a 3 ...) 
points, where 
a 2 d- 4a 2 d- 9a 3 ... = n 2 — 1, 
a 2 d- 3a 2 d- 6a 3 ... > \ (n 2 d- Sri) — 2, 
such that (for a curve of the order n passing once through each point a 2 , twice through 
each point oi 2 , ...) the number of conditions actually imposed on the curve is = or 
< £ (n 2 d- Sri) — 2. 
A system of (a 2 , a 2 ...) points such that the number of actually imposed conditions 
is less than a 2 d- 3or 2 d-..., may be termed a special system; we have, of course, the 
well-known case (ot 2 = w 2 ) of a system of ri points, such that any curve of the order n 
passing through ^ (n 2 d- 3?z) — 1 of these passes through all the remaining points {or what 
is the same thing, where the number of conditions actually imposed is =-|-(?i 2 d- 3?i) — 1}; 
and we have the following special system, which presented itself to Dr Clebsch, in 
his researches on the Abbildung of a quintic surface with two non-intersecting nodal