EEV. T. P. KIEKMAN ON THE K-PAETITIONS OE THE E-GON AND E-ACE. 253 
fj — 1 I 
of one of the r-gons of the division k—e\ is j .(e+1 ; for any e 
71 “• 3 
vertices may be chosen out of on one side of the monogonal axis. The whole 
number of operations described in Theorems P and S, by which a (l+^j-partitioned 
monogonal can arise, is the above operation performed on every (1-)-^— e)-partitioned 
monogonal in our register for every value of e, w, and h. The results, by Theorems E, 
P, and S, are all monogonally reversibles. And by Theorem Q we see that every 
(l-[-^)-partitioned (2/i+3)-ly reversible about loaded monogonal axes will be among 
these results, whatever be its nucleus ; and every singly reversible with a loaded mono- 
gonal axis will be among them ; for if this be cleared, we have before us one of our sub- 
jects of operation just referred to in our register. Wherefore 
and Er()-, 2»E“«'«"(>-, k), 
where the omission of the subscript n shows that all nuclei are included. This is a 
given number ; and as no monogonally reversible can be obtained by scoring any but a 
monogonal axis, it is the whole number k). 
LVI. Problem s. To find Po^(r, k), the mimher of {\-\-\)- 2 Mrtitioned x-gons singly 
remrsTble about a scored agonal axis. 
Ro^(r, ^’) and Ef (r, k) are obtained by the operations of Theorems P, E, S, by scoring 
both 2/i-ly and (2A+l)-ly reversibles. We write 
Ef(r, A:)=Er(r, k)'\ 
the first of the right member denoting the number of those r-gons obtained by scoring 
one of an odd number, and the second, those by one of an even number, of axes. 
The number Eo^(r, k)' is obtamed by Theorems P, Q, S, exactly as E™“(r, k) was 
obtained, except that the number of angles of the nucleus from which on one side of 
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the clear axis e scores can be di’awn, is in the first, instead of — ^ in the second. 
That is. 
where again the omission of the subscripts shows that constructions upon all nuclei are 
included. 
LYII. We have next to determine E“^(r, k)", obtained by scoring a clear axis out of 
an even number of axes (Theorem R). 
Whatever e may be, the constructions of Theorem R are enumerated by subtracting 
the symmetrical arrangements of the e scores from the whole number of arrangements, 
and di\dding the remainder by two ; for every unsymmetrical one occurs twice in this 
remainder, once as read from each end of the scored axis, which readings are the two 
configurations of the singly reversible about the axis (Theorem C). 
