384] 
ON THE TRANSFORMATION OF PLANE CURVES. 
7 
positions of the 2n — 2 points on the curve U = 0, and it is consequently a function 
of only the coefficients of the curve £7 = 0, being, as is obvious, an absolute invariant 
of the curve U — 0. 
26. And, moreover, if the curve JJ — 0 is by a transformation of the order n — 1, 
by means of 2n— 3 points on the curve as above, transformed into a cubic, then the 
absolute invariant 1 3 -r- J- of the quartic equation which determines the tangents to 
the cubic from any point 0' on the cubic (or, what is the same thing, the absolute 
invariant S 3 -f- T- of the cubic, taken with a proper numerical multiplier) is independent 
of the positions of the 2n — 3 points on the curve TJ = 0, being in fact equal to the 
above-mentioned absolute invariant of the curve U = 0. The like results apply to the 
transformation of the order n — 2. 
27. Suppose now that we have 2) > 2, and consider a curve of the order n with 
the deficiency 2), that is with \ (n 2 — 3n) — 2) + 1 dps, transformed by a transformation 
of the order n — 3 into a curve of the order 2) 4- 1 with deficiency 2); then, assuming 
the truth of the subsidiary theorem to be presently mentioned, it may be shown by 
very similar reasoning to that above employed, that the absolute invariants of the 
transformed curve of the order 2) +1 (the number of which is = 4D — 6), will be 
independent of the positions of the D — 3 points used in the transformation, and will 
be equal to absolute invariants( 1 ) of the given curve U = 0. 
28. The subsidiary theorem is as follows: consider a curve of the order 2) +1, 
with deficiency 2), that is, with \ 2) (Z) — 1) — 2) = |(2) 2 — 3D) dps; the number of 
tangents to the curve from any point O' on the curve is = (2) + 1) D — (2) 2 — 32)) — 2, 
= 4D — 2, (this assumes however, that the dps are proper dps, not cusps,) the pencil of 
tangents has 4D - 5 absolute invariants, and of these all but one, that is, 4i) — 6, 
absolute invariants of the pencil are independent of the position of the point 0' on 
the curve, and are respectively equal to absolute invariants of the curve. 
29. To establish it, I observe that a curve of the order D + 1 with deficiency 1), 
or with £ (D- — SB) dps, contains ^ (D + 1){D + 4) — |(i) 2 — 3D), =4>D + 2 arbitrary 
constants, and it may therefore be made to satisfy 4D+2 conditions. Now imagine a 
given pencil of 4D — 2 lines, and let a curve of the form in question be determined 
so as to pass through the centre of the pencil, and touch each of the 4D - 2 lines; 
the curve thus satisfies 4D — 1 conditions, and its equation will contain 42) + 2 — (42) - 1), = 3 
arbitrary constants. But if we have any particular curve satisfying the 42) — 1 conditions, 
then by transforming the whole figure homologously, taking the centre of the pencil 
as pole and any arbitrary line as axis of homology, so as to leave the pencil of lines 
unaltered (analytically if at the centre of the pencil ¿r = 0, y = 0, then by writing 
ax + (3ij + yz in place of z) the transformed curve still satisfies the 42) — 1 conditions, 
and we have by the homologous transformation introduced into its equation 3 arbitrary 
constants, that is, we have obtained the most general curve which satisfies the conditions 
in question. The absolute invariants of the general curve are independent of the 
1 It is right to notice that the absolute invariants spoken of here, and in what follows, are not in general 
rational ones.