486
[84
84.
ON THE DEVELOPABLE SUBFACES WHICH ARISE FROM
TWO SURFACES OF THE SECOND ORDER.
[From the Cambridge and Dublin Mathematical Journal, vol. v. (1850), pp. 46—57.]
Any two surfaces considered in relation to each other give rise to a curve of
intersection, or, as I shall term it, an Intersect and a circumscribed Developable 1 or
Envelope. The Intersect is of course the edge of regression of a certain Developable
which may be termed the Intersect-Developable, the Envelope has an edge of regression
which may be termed the Envelope-Curve. The order of the Intersect is the product
of the orders of the two surfaces, the class of the Envelope is the product of the
classes of the two surfaces. When neither the Intersect breaks up into curves of lower
order, nor the Envelope into Developables of lower class, the two surfaces are said to
form a proper system. In the case of two surfaces of the second order (and class)
the Intersect is of the fourth order and the Envelope of the fourth class. Every
proper system of two surfaces of the second order belongs to one of the following
three classes:—A. There is no contact between the surfaces; B. There is an ordinary
contact; C. There is a singular contact. Or the three classes may be distinguished
by reference to the conjugates (conjugate points or planes) of the system. A. The four
conjugates are all distinct; B. Two conjugates coincide; C. Three conjugates coincide.
To explain this it is necessary to remark that in the general case of two surfaces of
the second order not in contact (that is for systems of the class A) there is a certain
tetrahedron such that with respect to either of the surfaces (or more generally with
respect to any surface of the second order passing through the Intersect of the system
1 The term ‘Developable’ is used as a substantive, as the reciprocal to ‘Curve,’ which means curve of
double curvature. The same remark applies to the use of the term in the compound Intersect-Developable.
For the signification of the term ‘singular contact,’ employed lower down, see Mr Salmon’s memoir ‘On the
Classification of Curves of Double Curvature,’ [same volume] p. 32.
