of a Surface of the Second Order into itself. 333 



series passes through a point chosen at pleasure of the surface. 

 The lines belonging to the curves of the one series generate a 

 series of developables, the eMges of regression of which lie on 

 one of the surfaces intersecting the surface of the second order 

 in the four given lines ; the lines belonging to the curves of the 

 other series generate a series of developables, the edges of regres- 

 sion of which lie on the other of the surfaces intersecting the 

 surface in the four given lines ; the general nature of the system 

 may be understood by considering the system of normals of a 

 surface of the second order. Consider, now, the surface of the 

 second order as given, and also the two surfaces of the second 

 order intersecting it in the same four lines ; from any point of 

 the surface we may draw to the auxiliary surfaces four different 

 tangents ; but selecting any one of these, and considering the 

 other point in which it intersects the surface as the point corre- 

 sponding to the first-mentioned point, we may, as above, construct 

 the entire system of corresponding points, and then the line 

 joining any two corresponding points will be a tangent to the 

 two auxiliary surfaces 3 the system of tangents so obtained may 

 be called a system of congruent tangents. Now if we take upon 

 the surface three points such that the first and second are corre- 

 sponding points, and that the second and third are corresponding 

 points, then it is obvious that the third and first are correspond- 

 ing points ; — observe that the two auxiliary surfaces for express- 

 ing the correspondence between the first and second point, those 

 for the second and third point, and those for the third and first 

 point, meet the surface, the two auxiliary surfaces of each pair 

 in the same four lines, but that these systems of four lines are 

 different 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 -f- 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 mth side of the 

 polygon, then will the same property exist with respect to the 

 (m-j- 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 consecutive 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. 



