OF SIMULTANEOUS ORDINARY DIFFERENTIAL EQUATIONS. 553 
and the above equation for the inflexional tangents becomes 
0b, 
(dy, — c.dx) = dy — de) —=10- 
1 
Hence one of them is parallel to the plane y, = c,w. 
But it must lie in the tangent plane and pass through the point of contact. It is, 
therefore, the line 
Jy = Oe + dy, 
Yn = Cyt + ba, 
which was to be proved. 
It is remarkable that though from this point of view a congruency of inflexional 
tangents appears to be a particular kind of bitangential congruency, yet when they 
are considered from the point of view of the surface, the one is as general as the 
other, and every surface, whose degree is not 2 or 8, has one of each. 
Degenerate Inflecional Congruencies. (§ 40.) 
§ 40. An interesting question arises as to whether there is a degenerate form of the 
inflexional congruency when the surface it envelopes is replaced by a curve. In such 
a case the lines of the congruency that meet the curve at any one point will form a 
cone, and the cones belonging to consecutive points of the curve must not meet each 
other, for if they did they would envelope a surface, and the congruency would be of 
_ the bitangential kind. The only kind of conical surface that will meet the case is 
easily seen to consist of one or more planes touching the curve, and the congruency is 
made up as follows: Planes are drawn through the tangents to a curve according to 
some fixed law, and lines are drawn through the points of contact in each plane. The 
planes will envelope a torse on which the curve will lie, and thus the congruency may 
be said to consist of all the tangents to a surface at the points of a curve on that 
surface. 
The existence of these two kinds of congruency appears to have been overlooked in 
the classification given by Satmon (‘Geometry of Three Dimensions,’ § 453). It 
might also be desirable to break up the first category given there into two, the 
bitangents to a surface and the bitangents to a torse, that is, the ‘ lines in two planes” 
of a curve. The third category would then have to be divided into three, according 
as one, each, or neither, of the surfaces was developable, and the fourth into two. 
Consideration of the General Surface. (§§ 41-49.) 
§ 41. Take the surface 
Cl CH Oy OF) =O” phe eae cic gee a bly 
MDCCCXCV.—A. 4B 
