696 
Proceedings of the Royal Society 
twice, either in one or in two circles, and is read central in two 
triplets asj3, a s/3'. The four angles about s are asa opposite to 
fis/S', and asp' opposite to fisa. The crossing n occurs in circles 1 
and 4 in the triplets mnp and mnc 3 : its angles are 
mum opposite pnc 3 , and mne 3 opposite pnm, 
where the edges mn in the triplets are the two edges of a 2-gon mn. 
As mc 3 is an edge as well as mn and nc 3 , c 3 mn is a triangular mesh, 
and K = . . . mn ... is a mesh not triangular. Both are collateral with 
the 2-gon mn. 
The crossing p is read in two triplets, npa 3 and b 3 pa, of the first 
circle : its angles are 
npb 3 opposite a 3 pa, and npa opposite b 3 pa. 
The angles c 3 np (1) and npb 3 are in the mesh . . c 3 npb 3 . . = L, which 
since b 3 c 3 is an edge in circle 4, is the 4-gon e 3 npb 3 ~ L. Also the 
angles mnp (1) and npa (2) are in the mesh K = . . mnpa . ., which 
is collateral with the 2-gon mn and with the 4-gon L. 
The crossing a 3 occurs in the circles 1 and 4 in pa 3 b 3 and p 3 a 3 b 3 , 
where the two edges a 3 b 3 are different. Its angles are 
pa 3 p 3 opposite b 3 a 3 b 3 and pa 3 b 3 opposite p 3 a 3 b 3 . 
Here pa 3 p 3 and a 3 pa (2) are angles of the mesh M = . . . apa 3 p 3 . . 
At a in pab and bac 1 in circles 1 and 2 the angles are 
pab opposite bac x and pac 1 opposite bob. 
Here cyap, apa 3 (3), and p 3 ap 3 (3), are angles in M = . . c 1 apa 3 p 3 . ., 
which, since and d 1 p 3 are edges in circles 2 and 4, is the 6-gon 
M. = d 1 c 1 apa 3 p 3 . The angle pah is in the face K = . . . mnpab . . ., 
which is collateral with the 2-gon », the 4-gon L, the 6-gon M, 
and the 2-gon ab. 
If now we repeat at the four like-posited triplets of the first circle, 
cafm fmgg , ij 8 j , la n m, what w r e have done at the triplet pa 3 a, we 
shall complete the demonstration that our 15-gonal base 
K = abcdefghijklmnp 
is collateral with five 2-gons, five 4-gons, and five 6-gons. And 
such a 15-gon will be found in the same way from each of the 
