56 REV. J. P. KIRKMAN ON THE 
m e—2|1 m= et 
n(o+1) om a } 
Siro pe Gt ees ae 
ehenpte 
which is a portion of the number I(2v7+m+1,z) of singly 
irreversible 27-+m-+1)-edra having w triangles. 
R(2e+1,7) here stands for the whole number of singly 
reversible (27+ 1)-edra having ~ triangles, 7.e. 
R(2e+1,7)=R(Qe+ bk a ae + vii 
15. Prosiem fh. To Phos the ee a doubly 
reversible (4v-+m-+1)-edra having 22 triangles. 
No mention is made here of the subject to which these 
are reducible, because it must always be a doubly reversible, 
one of the number R?(4vw +1,22) ; for the lines 7, which dis- 
appear (Art. 3) in such reduction, are placed symmetrically 
about two axial planes: wherefore that symmetry must 
remain in the subject P to which the figures reduce them- 
selves. 
One of the axial planes (Art. 5) of P must contain £’, 
one of the 47-38 edges k, which x’ (Art. 13) can receive no 
point 7, if the result is to be doubly reversible and all the 
summits triaces. 
In any one of the quadrants between these planes will be 
seen 2-1 edges k, on which 47 points 7 are to be planted, 
and the configurations are to be completed so that every 
quadrant shall read as the reverse of that on either side of 
it. The number of results will be, 
EONS 
™ 
eee ee Been “eyo 
Pen 5 
ar 
wherefore eG 
» G+) 
R?(den +m + 1,20) = 24. Faq —— RX(4e + 1,22), 
which is the whole number R*(4v + m+ 1,22.) 
This vanishes if e=1, unless m=O, showing that the 
wedge is the only doubly reversible with two triangles. 
ee 
