EEV. T. P. KIKKMAN ON AUTOPOLAE POLYEDKA. 
191 
X. Thirdly, let 
E,-e=E;-/+, 
or 
2-2e=2-e'-f±, 
which gives 
2e=e'-^f'+ 
2/=/+e'+. 
The only interpretation of this, consistent with e'f not ef, is 
2e=d-\-f' 
which can be true only when 
2e—2f=+lcr; 
but 
and 
2e>2r, hence k<2, 
2e-2f=±r, 
or 
e=f±¥^ 
is the only possible relation ; i. e. ef is the diameter of an even-angled base. 
The equation 2e=e'-\-f' is 2e=2e'+^, giving 
Therefore /•=4m. 
By this we see that, when /’=4m, the diameters all pair themselves m summits apart 
into generators ef and ef\ of the sum (r+2)-edron ; for no value of e' and/"' can coincide 
with e and y from which it is obtained, since 
along with 
^=e+^r 
and 
cannot coexist. 
XI. Lastly, let 
f=e±lr 
E,-6=^-F,+ , 
or 
2-2e=e’+f-l±; 
whence comes 
2e-\-e’ 
and 
2/+/' +e'=3 + . 
Of these, the only meaning that consists with ef’, different from ef is 
2e-\-e'-\-fz=z3-{-kr 
2/+/'+e'=3+Fr, 
2(e—f)=(k—^)r. 
