112 
ON THE HOMOGRAPHIC TRANSFORMATION OF A SURFACE &C. [122 
for the different pairs of auxiliary surfaces. The same thing of course applies to any 
number of corresponding points. We have thus, finally, the theorem, if there be a 
polygon of (m +1) sides inscribed in a surface of the second order, and the first side 
of the polygon constantly touches two surfaces of the second order, each of them 
intersecting the surface of the second order in the same four lines (and the side 
belong always to the same system of congruent tangents), and if the same property 
exists with respect to the second, third, &c.... and with side of the polygon, then will 
the same property exist with respect to the (m + l)th side of the polygon. 
We may add, that, instead of satisfying the conditions of the theorem, any two 
consecutive sides of the polygon, or the sides forming any number of pairs of con 
secutive sides, may pass each through a fixed point. This is of course only a 
particular case of the improper transformation of a surface of a second order into 
itself, a question which is not discussed in the present paper.