EEV. T. P. KIEKMAI^ ON AUTOPOLAE POLTEDEA. 
199 
(1, 5 — ^) or (2, 5+1^). In the first case, 1 is a 4-ace and 2 a 4-lateral; in the second, 
2 is a 4-ace and 1 a 4-lateraL In either case 1 and 2 are 4-aces, but in both (12) is an 
edge of the triangle 5, and therefore not convanescible. Therefore s has no existence 
when r=5. Let, next, .2^=4 and y=3; then r=4. 
We are to look in s made from the 5-edral pyramid for a convanescent s not passing 
through 0 or o'. The simplest way of doing this is to write out the four results of draw- 
ing the four generators ?, in the 4-gonal Q. The eight edges of the pyramid are repre- 
sented thus : 
1o3i343s 
40833234 
4o343i3i 
3 i 34324 o ^2333343 33323440 
343i3i4o5 
where the first and third places in the quadruplet show the rank and signature of the 
left and right summits, and the second and fourth, those of the upper and lower faces of 
the edge ; and each is written over its gamic. 
If we draw 12^ to the mid-point of the edge 23, the face 3 becomes a 4-gon, the 
summit 3 a 4-ace, and we have the 4-gon 1 and the 4-ace I. The 4o of the base and 
summit becomes 4o and 3op and the new edges 00 ' and its gamic appear; while instead 
of 23, the second edge and its gamic, we have 22^ and 2^3 with thefr gamics. The 
result is 
3o, 41^482 
^ 0 , 824332 , 
4082,4343 
40433284 
4o844i4i 
^0,82,4o4i^ 
(12.) 
4i34323o, 
^ 24332 , 3o, 
^ 2,434340 
4382^440 
^44i4i4o 
^2,4o4i3o,J 
The results of drawing 13^ to the mid-point of 34, 23^ to the middle of 34, and 24^ to the 
middle of 14, 
are in order following : — 
4o4i3442 
4o423333 
40834233 , 
^ 0 , 83,4284 
3o,844i4i 
^0,4i4o83,^ 
(13.) 
4i34424o 
42833340 
^342^3,40 
^3,42^430, 
344i4i3o, 
4i4o33,3o,J 
4o3i3452 
^0,62^383 
^ 0 , 83 ^ 233 , 
4083,5284 
4o843i3i 
4062 ^ 0 , 83 ,^ 
(23.) 
3i34524o 
52833330 , 
^352^3,30, 
^3,52^440 
^48i3i4o 
^ 280 ,^ 3,40 J 
3o4i^442 
4o423333 
40834234 
4o844i34' 
3o,84,4i4i 
^0,424o84,^ 
(24.) 
4i34423o, 
42833340 
^342344o 
344i344o 
34,4i4i3o, 
424o34,8o,J 
In all these, the only convanescible edges, not through 0 or Op are 32,43434o ^^4 
344 i4i4o ill (12 J, either of which is in two 4-laterals, and the summits of neither are 
in two faces having any other common edge. Either therefore is convanescible ; but 
with their present signatures they will neither of them vanish so as to give a 5-ace by 
the union of their summits ; nor is either of them a generator introduced into the 5-gonal 
Q, in our construction of the (r+2)-edra P. For they are both nodal gamics ; the first 
meeting its gamic at 3, and the second at 1, while the generators of the class P never 
met their gamics. The first edge cannot vanish without the loss of two edges at the 
