60 
MR. H. M. TAYLOR ON A SPECIAL FORM OF THE 
The results of §12 are all to be found in Sturm, ‘ Synthetisclie Untersuchungen 
liber Flachen Dritter Ordnung.’ 
§ 13. It is well known that the number of triangles on a cubic surface is 45. 
We may calculate the number of closed quadrilaterals, pentagons, and hexagons, 
restricting the denomination to polygons of the proper number of sides, no two sides 
of which intersect each other exce]:)t consecutive sides. 
By inspection of one of the figures (and for this purpose Figure D is the most 
convenient) it is easy to see that the number of lines on the surface Avhich intersect 
both lines of an open angle is. 
both the end lines and no otliers of an open trilateral is . 
,, ,, ,, ,, Cjuadrilateral is. 
,, ,, ,, ,, quinquilateral is 
1 (see lines 14, 24), 
4 (see lines 25, 14, 
24), 
3 (see lines 19, 12, 
25, 14), 
1 (see lines 17, 1,19, 
12, 25), 
and that the number of lines on the surface which intersect only one line, and that, 
a specified end line of an open angle, is.8 
only one line (an end line) of an o^^en trilateral, is ... 4 
„ ,, ,, quadrilateral, is . . 1 
„ ,, ,, quinquilateral, is . 0, 
and that the number of lines on the surface which intersect none of the lines 
of an open angle, is.8 
,, trilateral, is .... 4 
,, quadrilateral, is ... 3 
,, quinquilateral, is . . 3. 
(An open sexilateral does not exist on the surface.) 
By means of Figure D we can see, by inspection of the lines 8, 2, 3, 10, that they 
form a closed quadrilateral, and that some one of them is intersected by every other 
line except 15. 
By inspection of the lines 13, 18, 8, 2, 3, that they form a closed pentagon, and 
that some one of them is intersected by every other line except 11 and 16, ■which do 
not intersect. 
By inspection of the lines 2, 3, 10, 11, 9, 4, that they form a closed hexagon, and 
that some one of them is intersected by every other line except 15, 18, 19, which do 
not intersect. 
