8 
ON THE TRANSFORMATION OF PLANE CURVES. 
[384 
3 arbitrary constants introduced by the homologous transformation; and they are 
consequently functions of only the coefficients of the given pencil of 42) — 2 lines; 
this being so, it is obvious that they will be respectively equal to absolute invariants 
of the pencil of 42) — 2 lines. The number of the absolute invariants of the general 
curve of the order D + 1 is = £ (2) + 1) (2) + 4) + 1 — 9, but there is a reduction = 1, for 
each of the dps, hence in the present case the number is ^ (2) + 1) (2) + 4) — \ (2) 2 — 32)) — 8, 
= 42) — 6; and there are thus 42) — 6 absolute invariants of the curve, each of them 
equal to an absolute invariant of the pencil; that is, of the 42) — 5 absolute invariants 
of the pencil, there are 42) — 6, each of them equal to an absolute invariant of the 
curve, and consequently independent of the position of the point 0' on the curve; 
which is the theorem which was to be proved. I believe the reasoning is quite 
correct, but there are some points in it which require further examination, it is 
therefore given subject to any correction which may hereafter appear to be necessary. 
30. The general subject may be illustrated by considerations belonging to solid 
geometry. If we imagine the original curve and the transformed curve as situate in 
different planes, then joining each point of the original curve with the corresponding- 
point on the transformed curve, we have a series of lines forming a scroll (skew 
surface): if the two curves are of the orders n, n' respectively, then the complete 
section by the plane of the original curve is made up of this curve of the order n, 
and of n' generating lines; and similarly the complete section by the plane of the 
transformed curve is made up of this curve of the order n\ and of n generating 
lines. Conversely, given a scroll of the order n + n\ any two sections of this scroll, 
being in general curves of the same order n + n, are rational transformations the one 
of the other; but for the general scroll of the order n + n', it is not possible to find 
sections breaking up as above.