447] ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. 199 
also each passing through the same four given points; the second set of equations is 
x : у : z — X' : Y' : Z\ where X' = 0, Y' = 0, Z'= 0 are like cubics in the second plane. 
27. Generally suppose that the ri 2 — 1 points in the first plane are made up of 
ctj points, which are simple points; a 2 points, which are double points; a 3 points, which 
are triple points,... a n _ 1 points, which are (n — l)tuple points = 1 or 0), on each of 
the three curves; these will represent a system of n 2 — 1 points if only 
a x + 4ot 2 + 9a 3 ... + (n — l) 2 a n _ x = n 2 — 1. 
The number of conditions imposed on each of the curves X = 0, Y = 0, Z = 0 will be 
oti + 3a 2 + 6a s ... + \n (n — 1) ot n _ x ; for the reason presently appearing, I exclude the case 
of this being <^n(n + 3) — 2 ; and therefore assume it to be = \n (n + 3) — 2. In fact, 
writing 
+ 3ot 2 + 6a 3 ... + \n (n — 1) a n _i = ^n(n + 3) — 2, 
this combined with the former equation gives 
a 2 + 3a 3 ... + | (n - 1) (n - 2) a n _ x = J (и - 1) (n - 2) ; 
viz., the singularities are equivalent to ^ (n — 1) (n — 2) double points, that is, to the 
maximum number of double points of a curve of the order n; or say each of the 
curves X = 0, F= 0, Z = 0 is a curve of the order n having a deficiency = 0 ; that is, 
it is a unicursal curve of the order n. Hence also, taking (a, b, c) any constant factors 
whatever, the curve aX + bY+cZ= 0 is unicursal. 
28. It is important to remark that the conclusion follows directly from the general 
notion of the rational transformation; in fact, the equation aX + bY+cZ= 0 is satisfied 
if x : у : z = X' : Y' : Z'; ax' + by' + cz' = 0. The last of these equations determines 
the ratios x' : y' : z' in terms of a single parameter (e.g. the ratio x' : y'), and we 
have then x : у : z expressed as rational functions of this parameter; that is, the curve 
is unicursal. 
29. Suppose for a moment that it was possible to have 
Я] -)- Зое о -)- 6х 3 ...-(- ^¡11 (ух 1) 1 ^ ^ хъ (уъ 3) 2. 
Combining in the same way with the first equation, it would follow that 
a 2 + 3a 3 ... + (n — 1) (yi — 2) oe n _ x > (u 1) (yi — 2), 
which would imply that the curves X = 0, Y = 0, Z = 0 break up each of them into 
inferior curves: but more than this, the coefficients a, b, c being arbitrary, it would 
imply that the curve aX + bY + cZ=0 breaks up into inferior curves; this can only be 
the case if X, Y, Z have a common factor, say M; that is, if X, Y, Z — MX ly MY 1} MZy. 
but we could then omit the common factor, and in place of x' : y' : z — X : Y : Z 
write x : у : z' = X x : Y x : Z x , where X x = 0, F^O, Z x = 0, are proper curves, not 
breaking up ; the above supposition may therefore be excluded from consideration.