EEV. T. P. KIEKMAJ^ ON ATJTOPOLAE POLTEDEA. 
215 
Hence the only vanescible pairs are the second and third. As they stand, the summits 
read thus with their signatures, 
The vanescence of the third pair gives us the reading 
^ 1 ^ 27 ^ 3 ^ 4 '^ 5 ^ 6^8 » 
and now the fourth pair has become the only vanescible one, giving the reading 
^ i 427^34:48^5^6» 
which by the vanescence of the sixth pair becomes 
^1^2 7^3^48^56? 
the pyramid on the base 1234. 
If we had begun by making the second pair vanish, 
^ 1 424344^45 3 g 37 3 g 
would have reduced to 
^1^2^86^4^537^85 
for the third summit losing both the edges of the nodal angle, 36 and 32, can unite with 
the sixth to form a 4-ace only. The sixth pair vanishing, gives us 
3 i 42 8333443537, , 
which reduces, as before, to the pyramid on the base 1234 by the vanescence of the fifth 
pair. 
The vanescible paus of the third 8-edron are the second and the fifth, which give 
us two ways of reducing it to the pyramid on the base 1234. Or we can reduce it 
by the evanescence of 354482385 4:44i 36485 and convanescence of 3^41 3658 and 
3342 344:8 fo pyramid on pentagonal base. 
XL. All the autopolars above given can be represented by square paradigms, showing 
all the faces, summits, edges and angles of the figiue ; for a closed polygon can be drawn 
through the summits of any of them. For example, the three last written have through 
then* summits the circles 15238746, 15672384, 15238746 ; «. e. the duadsof these circles 
occur among the non-contiguous duads of the subindices. If these circles be employed 
as du’ected in my paper “On the Representation of Polyedra,” in the volume for 1856 
of the Transactions of the Royal Society, the paradigms are easily written out. But I 
do not find that any mode of representation is so simple and distinguishing as that above 
given in art. XXXVI. and those following it. 
The figures in Plates XII., XIII., XIV., are intended to illustrate the Articles of this 
Memoir to which reference is made in them. 
