EEV. T. P. KIEKMAN ON THE K-PAETITIONS OF THE E-GON AND E-ACE. 249 
Avith an agonally reversible (about one or more axes) (l-|-go)"P8'i’titioned (4-|-2ao)-gon. 
The intervals of sides Avill be loaded as in the last preceding construction, to satisfy 
r—n- 
!''1~ A(2-|-2c4())-|-2A^<Zj-f-652~|“ . . 
^ = ~f~^2 ~f~ • • H~ 
(a^>0, e^:i>a^—l), (n—2m+l), 
(D'.) 
(4-l-2ao) and being numbers selected at pleasure from our register of agonally or ago- 
diagonally reversibles. Putting, as in XIX., 
2^{2i>™-.(4+2ao, g-l-R^-^<^>(4+2ao, eo)}=H, 
and forming the product from a solution of (D'.) which contains those numbers 
we deduce, as before, that 2H . A^, comprising as many products under 2 as those 
2h 
solutions of (D'.), all orders of the quantities e^, See. being counted as solutions, is the 
total number of axial configurations of (l+^)-partitioned r-gons having h loaded mono- 
gonal axes and built on the w-gonal nucleus when n is odd. 
And as one of these configurations is seen on each r-gon of the number u,(^ 2 i+i)A.mo^y.^ 
and only one, read from the loaded end of the axis, we obtain 
and 
^)„=2H. A«-a— k)„, 
2k 
where n is odd, and ^■>0. 
L. Now let the h axes be agonal axes of the %-gon; ^. e. let n be even. 
One end of eveiy axis will carry (Theorem O) the same (3-l-2a„)-gon (1 -fsoj-partitioned 
and reversible about at least one monogonal axis. The other ends will carry the same 
j2-f2a„J-gon, -partitioned and reversible about at least one agonal axis, which 
Tt **“ 
will be no load if an=0. The interval of - sides between these reversibles will be 
2h 
loaded, on the mth side from the first, with a (l+e„j)-partitioned (2d-«,„)-gon; and the 
equations to be satisfied are. 
r = CL n-^h \-\-hcCn ,^1 
V 2A / 2A 1 
^ = W“|“/i£o •\-21ll6i-\~ 62 ~\~ . • '\~€n—^h\-\~llZ^y J 
V ) 2A 
(D".) 
where a„,>0, 1, and Ko, 2 ^ 9,re chosen at pleasure, the first pair from our 
2A 2A 
register of monogonally reversibles, and the second from our agonally or ago-diagonally 
reversibles. 
Taking now any one solution of these equations, the product H"MA of 
H 
"=2„/2R'"-“^/2+a„, 2 AI, 
I \ 2A W \ 2A 2A/ / 
2 L 
MDCCCLVII. 
