PARTITIONS OF THE X-ACE AND THE X-GON. 57 
16. Prostem #. To determine the number of doubly 
irreversible (4v+m-+1)-edra reducible to doubly rever- 
sible (4s -+1)-edra having 22 triangles. 
Nothing prevents us here from depositing points 7 on 
the axial edge # of the subject, inasmuch as the opera- 
tions in one half of that edge can be repeated in the other 
half without introducing tessaraces. 
On one side of the axial plane which bisects k’ we see 
2-1 edges k, including the half of #’, wherefore the 
number of ways of drawing 4m lines 7 is 
ene 
[== lanai & 2 ? 
wf 
which operations are to be repeated in direct order on the 
other side of the plane. This gives us only doubly irre- 
versible results, but every one is twice obtained because 
of the reversible configuration on which we have operated. 
To every face h on one side of our bisecting plane corre- 
sponds another face h’ on the same side, such that the 
sequence read from / in one direction is identical with 
that read from /’ in the opposite; so that every sequence 
is twice constructed. We have therefore to divide the 
above by two; hence 
2¢—2 41 
(sith) m=2 
PR(4¢+m+1,22)= eC) Oe »R?(4e 41,22) ; 
which is a portion of the number ?’(4w+m-+ 1,22). 
No new results can be obtained by operating on the semi- 
solid cut off by the other bisecting plane; for if in P’ a 
result so gained we move the bisecting plane to its old 
position in our subject, we sce on one side of it a configu- 
ration before enumerated, since all possible ones have been 
before enumerated. 
17. Prostem 7. To determine the singly reversible (4a 
VOL. XV. I 
