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-