392
ON THE TRANSFORMATION OF UNICURSAL SURFACES.
[523
The necessity for the term co appears by the consideration that if we apply to
the plane figure a Cremona-transformation, thus obtaining a new transformation of the
surface, the value of 2a,. will in general be altered ; whereas the expressions for q, t
should it is clear remain unaltered; and it arises as follows, viz. for certain transfor
mations of the surface the curve of the order (k — n 4- 4) n — 3, passing (k — n -t- 4) r — 1
times through each point a r and assumed to be the projection of the residue, is not
an indecomposable curve but contains a certain number co of factors (each belonging
to a unicursal curve definable by means of the number of its passages through the
several points a r ), which factors are to be rejected in order to obtain the equation of
the proper residue. Thus reverting to the transformation
x : y : z : w = x'* : y'* : : ix' + y' + /) 2
of Steiner’s surface, the projection of the quadric residue was (as already remarked) a
line; applying to the plane figure the ordinary quadric (or inverse) transformation we
introduce three fixed points, (a 2 =3), say these are A, B, G\ viz. in the new trans
formation of the surface the projection of any plane section is a quartic curve having
a node at each of the fixed points: the projection of the residue ought clearly to be
a conic through the three points; but according to the general formula it is a quintic
having at each of these points a triple point: the quintic is in fact made up of the
lines BG, GA, AB and of the conic which is the proper residue; viz. in the case in
question there are 3 factors thrown out, or we have co = 3. To apply this to the
second investigation of 2q + 91, by comparison of the two deficiencies, observe that in
general if a curve is made up of co +1 indecomposable curves, the deficiency of the
compound curve is equal to the sum of the deficiencies of the component curves — co;
hence if co of the curves are unicursal, the deficiency of the compound curve is equal
to that of the remaining curve — co ; or, what is the same thing, the deficiency of the
remaining curve is = that of the compound curve + co; and the addition of the term
+ to to the expression for the deficiency is thus accounted for. It is easy to see that
a like explanation applies to the first investigation of 2q + 91.
I further remark, reverting to the equations
x : y : z : w = X' : Y' : Z' : W'
of the transformation, that the product of the co factors is given as the common factor
(if any) of the Jacobians
J(Y\ Z', W'), J(Z', W', X'), J{W, X\ Y') and J(X', Y', Z').
Such common factor exists whenever we can by a Cremona-transformation of the plane
figure reduce the number of the points a r upon which the transformation of the
surface depends; viz. for any given transformation of the surface, co is equal to the
excess of above the minimum value of Xa r , or, what is the same thing, 2a,. — co
is equal to the minimum value of 2a r , and is thus independent of the particular trans
formation. And of course if 2a r has this minimum value, viz. if the transformation
is such that the number of the points a r cannot be reduced by any Cremona-trans-
