S8 
MR. H. M. TAYLOR OR A SPECIAL FORM OF THE 
KLMN = (T - K) (T - L) (T - M) (T - N), 
where K, L, M, N, T equated to zero represent planes. 
In §§ 2-9, it is shown how to obtain the equations of the twenty-seven lines on the 
surface whose equation is 
xyzu = {x — aT) [y — 6T) {z — cT) {u — dT), 
and further it is shown which of the twenty-seven lines intersect each other. 
In § 10 the method of representation by a plane-diagram is explained, and the 
remaining part of the paper consists chiefly in deducing mutual relations between the 
lines by means of the diagram or one of its transformations. 
It may be explained that of the four transformations of the diagram, Figure A is 
arranged to show that the lines which are numbered 1 to 15 form in threes, five 
triangles; the remaining 12 lines, which are numbered 16 to 27, do not form a 
single triangle by themselves."^ 
Figure B is arranged to show that not only can nine planes be drawn to pass 
through all the twenty-seven lines, but that they can be arranged in three sets of 
nine each, such that each set forms three triangles in two distinct ways. 
Figure C is arranged to exhibit what is called a “ double six ” in the left hand top 
corner. It is of use for observing wFat lines intersect or do not intersect a number 
of non-intersecting straight lines, such as the six numbei’ed 20, 21, 8, 11, 3, 4, or the 
six numbered 26, 27, 5, 2, 9, 10. 
Figure D is arranged to show that it is possible to form a closed polygon of all the 
twenty-seven lines, such that no side intersects either of the sides next but one to 
itself. 
This figure is of use for observing what lines intersect, or do not intersect, the 
sides of a closed quadrilateral, pentagon, or hexagon, such as are formed by the 
lines numbered 26, 17, 1, 19 ; 16, 23, 26, 17, 1, and 2, 3, 10, 11, 9, 4 resiaectively. 
* Ifc has been remarked as an omission in this paper that the fact that these twelve lines form a 
double six ” is nowhere stated. 
