G8 
MR. H. M. TAYLOR A SPECIAL FORM OF THE 
This we can complete in three distinct ways by filling up the corner spaces, so that 
the rows and the columns will all give triple tangent planes. 
The figures are as follows :— 
!) . 7 8 
12 3. 
4 5 G 
11 12 . 10 
and 
1G . 7 23 
12 3. 
4 5 G 
19 12 . 24 
17 . 7 22 
12 3. 
4 5 G 
18 12 . 25 
This proves that from every closed quadrilateral we can obtain three distinct pairs 
of tetrahedrons in perspective, and, therefore, three distinct perspective planes. 
If we have two triple tangent planes which do not possess a line in common, say, 
1 2 3 
we can obtain from them in nine difterent ways a pair of tetrahedrons in perspective, 
and from every pair of such tetrahedrons we can obtain 4.3 = 12 pairs of such triple 
t.'ingent planes. 
Therefore, the number of perspective planes 
= the number of pairs of tetrahedrons in perspective, 
= “A X the number of such pairs of triple tangent planes, 
. A4.JL4 = 3 . 45 . 4 = 6 . 90 = 540. 
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