84] 
ON THE DEVELOPABLE SUBFACES &C. 
487 
or inscribed in the Envelope) the angles and faces of the tetrahedron are reciprocals 
of each other, each angle of its opposite face, and vice versa. The angles of the 
tetrahedron are termed the conjugate points of the system, and the faces of the 
tetrahedron are termed the conjugate planes of the system, and the term conjugates 
may be used to denote indifferently either the conjugate planes or the conjugate points. 
A conjugate plane and the conjugate point reciprocal to it are said to correspond to 
each other. Each conjugate point is evidently the point of intersection of the three 
conjugate planes to which it does not correspond, and in like manner each conjugate 
plane is the plane through the three conjugate points to which it does not correspond. 
In the case of a system belonging to the class B, two conjugate points coincide 
together in the point of contact forming what may be termed a double conjugate 
point, and in like manner two conjugate planes coincide in the plane of contact (that is 
the tangent plane through the point of contact) forming what may be termed a double 
conjugate plane. The remaining conjugate points and planes may be distinguished as 
single conjugate points and single conjugate planes. It is clear that the double conju 
gate plane passes through the three conjugate points, and that the double conjugate 
point is the point of intersection of the three conjugate planes: moreover each single 
conjugate plane passes through the single conjugate point to which it does not corre 
spond and the double conjugate point; and each single conjugate point lies on the 
line of intersection of the single conjugate plane to which it does not correspond and 
the double conjugate plane. 
In the case of a system belonging to the class (G), three conjugate points coincide 
together in the point of contact forming what may be termed a triple conjugate point, 
and three conjugate planes coincide together in the plane of contact forming a triple 
conjugate plane. The remaining conjugate point and conjugate plane may be distin 
guished as the single conjugate point and single conjugate plane. The triple conjugate 
plane passes through the two conjugate points and the triple conjugate point lies on 
the line of intersection of the two conjugate planes; the single conjugate plane passes 
through the triple conjugate point and the single conjugate point lies on the triple 
conj ugate plane. 
Suppose now that it is required to find the Intersect-Developable of two surfaces 
of the second order. If the equations of the surfaces be T = 0, T'=0 (T, T' being 
homogeneous functions of the second order of the coordinates f, 77, £, <o), and x, y, z, w 
represent the coordinates of a point upon the required developable surface: if more 
over U, U' are the same functions of x, y, z, w that T, T' are of £, 77, £, and 
X, Y, Z, W; X', Y', Z', W denote the differential coefficients of U, U' with respect 
to x, y, z, w; then it is easy to see that the equation of the Intersect-Developable 
is obtained by eliminating 77, £, co between the equations 
T = 0, T' = 0, 
X%+Yr) + Z£+ IF &> = 0, 
X'£+ Y'y + Z'£ + W'eo = 0.