EEV. T. P. EIEKMAJf ON THE K-PAETITIONS OF THE E-GON AND E-ACE. 251 
form of equations (D".), made with numbers s#, selected from our register, as 
2h 2h 
above stated (L.). 
~mo 
From this value of Eo (r, we can proceed to find, when n is even, Ro”*“(r, k)^ for 
all values of ^>1, which make 
w, /•— A^l+2a„+a^^, and k—1i\zQ-\-z^, 
divisible by 2A, But we cannot find E™"(y, k)^ by these formulae, there being no such 
division reducible to a nucleus (VIII.). 
Thus we have found (XLIX., L.) RJ*"® (r, k)n for any value of w>2. 
LII. AR the numbers R of article XX. have now been determined, for every value 
of h when the axes are not all loaded, and for every value of A>I when they are all 
loaded, the nucleus being 7i-gonal, and n<^^. We have next to enumerate the doubly 
and singly reversibles which are buRt on the 2-gonal nucleus, that is, upon a line, which 
is of course a drawn axis of reversion of the r-gon, and also the singly reversibles which 
have an e-scored axis. These have no proper nucleus (Theorem K). 
These r-gons now to be discussed fall into the classes following, 
R2agdi(y., k)^, R*(r, k\, Er(^5 ; 
of which the first two have a drawn diagonal axis standing as a perpendicular score upon 
an undrawn axis (Theorem G) ; the third has a single drawn diagonal axis (Theorem G) ; 
and the fourth, fifth and sixth have a single undra'wn and scored axis of reversion (Theo- 
rem K). The subscript 2 in the first three symbols shows that the nucleus is a drawn 
line, which may be considered as a 2-gon reversible about a diagonal and an agonal axis. 
Problem o. To find R^*®'“(r, k) 2 , the (1 -\-Y)^artitioned x-gons built on a nucleus-line to 
have a diagonal and an agonal axis of reversion. 
If the drawn diagonal be erased, the figure will be still reversible about the same two 
axes undrawn, for (Theorem B, VI.) the erasure has not disturbed the symmetry about 
either axis. But the erasure may have restored the symmetry about some w-gonal 
nucleus of which that erased line is a clear diagonal axis (Theorem R, XVIII.). And 
every r-gon which has a clear diagonal axis perpendicular to an agonal one, that is, every 
one of k—V)n and of ^— !)»? 'whatever h or n may be, if /i>2 
{Cor. 1, VII.), will give us one and only one (Theorem G) of R^“^*'(r, k)^.> by the drawing 
of any diagonal axis of reversion; for all these axes bisect the same configuration. 
Wherefore 
R^“^"'(r, ^)2=2,2:„{Rf ^-l)„+R't"+^>“^"'(r, A:-I)J, 
including all values of h, and of n>‘l in our register; n of course being even, else it 
could have no diameter. 
As we have nothing in our register imder Jc)ni R^''^*(8, ^)2=0, whatever k 
may be. 
LIII. Problem p. To find R^^‘(r, k) 2 , the number of {1 -\-\L)-fartitioned r-gons built on 
a line-nucleus, to have two diagonal awes of reversion. 
2 L 2 
