4 
ON THE TRANSFORMATION OF PLANE CURVES. 
[384 
expressible rationally in terms of (x : y : z)\ it is convenient to consider the trans 
formation from this point of view, and I now proceed to the independent development 
of the theory, as follows: 
15. We have a given curve U=(x, y, z) n = 0, with deficiency D, which is by the 
transformation £ : rj : £ = P : Q : R (where P, Q, R are given functions (x, y, z) k each 
of the same order k) transformed into the curve T = (£, r/, £)" = 0. The transformed 
curve has, as we know, the same deficiency D as the original curve. 
16. To find the order of the transformed curve, we must find the number of 
its intersections with an arbitrary line a% + brj + c£= 0. Writing in this equation 
£ : 7] : £ = P : Q : R, we obtain the equation aP + bQ + cR = 0, and combining there 
with the equation U = 0, the two equations, being of the orders k and n respectively, 
give kn systems of values of (x : y : z), and to each of these, in virtue of the 
equations £ : 77 : £ = P : Q : R, there corresponds a single set of values of (£ : rj : £), 
and therefore a single point of intersection ; hence the number of intersections, that is, 
the order of the transformed curve, is =kn. 
17. If, however, the curves P = 0, Q = 0, R = 0, meet in an ordinary point of 
the curve U = 0, then it is easy to see there is a reduction =1 in the foregoing 
value; and so if they meet in a dp of the curve U = 0, then there is a reduction 
= 2. More generally if the curves P = 0, Q = 0, R= 0 each pass through the same a 
dps and /3 ordinary points of the curve U = 0, then there is a reduction = 2a + /3. In 
fact the curve aP + bQ + cR = 0, meets the curve U = 0, in kn points; but among 
these are included the a dps, each counting as 2 intersections, and the /3 points; the 
number of the remaining intersections is = kn —2a — /3, and the order of the trans 
formed curve is equal to this number. 
I assume that we have k< n: 
18. A curve of the order k may be made to pass through £k (k + 3) points; it 
is moreover known that if any three curves, P = 0, Q= 0, R = 0, of the order k each 
pass through the same \k (k + 3) — 1 points, then the three curves have all their inter 
sections common, the equations being, in fact, connected by an identical relation of 
the form aP + ¡3Q + 7R =0. To make the order of the transformed curve as low as 
possible, we must make the curves P = 0, Q = 0, R = 0, meet on the curve U — 0 in 
as many points as possible, and it appears from the remark just made, that the 
greatest possible number is =£&(& + 3) — 2; in particular, for k = n— 1, n — 2, n — 3, 
the number of points on the curve U = 0 will be at most equal to ^(n 2 +n)— 3, 
^ (n 2 — n) — 3, ^ (a 2 — 311) — 2, respectively. 
19. Hence, considering the curve U =0 with deficiency P, or with ^ (n 2 — 3a) — D + 1 
dps, first if k = n— 1, we may assume that the transforming curves P = 0, Q = 0, R = 0 
of the order n— 1, each pass 
through the £ (n 2 — 3n) — D + 1 dps, 
and through 2n + D — 4 other points, 
together £ (n 2 + n) — 3 points of the curve U = 0.