384] 
ON THE TRANSFORMATION OF PLANE CURVES. 
5 
I his being so, each of the three curves will meet the curve U — 0 
in the dps, counting as n 2 - Sn - 2D + 2 points, 
in the 2 n + D — 4 points, 
an d in D + 2 other points, 
together n 2 — n points; 
whence the order of the transformed curve is = D + 2. 
20. In precisely the same manner, secondly, if k — n — 2, then we may assume 
that the transforming curves P = 0, Q = 0, R = 0, of the order n - 2, each pass 
through the \ (n 2 — Sn) — D + 1 dps, 
and through n + D — 4 other points, 
together \ (n? — n) — 3 points of the curve U = 0 ; 
and this being so, each of the three curves will meet the curve TJ = 0 
in the dps counting as n 2 — Sn — 2D + 2 points, 
in the n + D — 4 points, 
and in D + 2 other points, 
together n 2 — 2n points; 
whence the order of the transformed curve is also in this case = D + 2. 
21. I was under the impression that the order of the transformed curve could 
not be reduced below D + 2, but it was remarked to me by Dr Clebsch, that in the 
case D > 2, the order might be reduced to D + 1. In fact, considering, thirdly, the 
case k = n— 3, we see that the transforming curves P — 0, Q = 0, R = 0 of the order 
n — 3 may be made to pass 
through the ^ (n 2 — Sn) — D + 1 dps, 
and through D — 3 other points, 
together ^ (n 2 — Sn) — 2 points of the curve U = 0 ; 
and this being so, each of the three curves meets the curve U= 0, 
in the dps counting as (n 2 - Sn) — 2D+ 2 points, 
in the D— 3 points, 
and in D + l other points, 
together n 2 - Sn points; 
whence the order of the transformed curve is in this case =Z)+1. 
22. The general theorem thus is that a curve of the order n with deficiency D, 
can be, by a transformation of the order n-1 or n- 2, transformed into a curve of