§ 2. XLI.-XLIII.] EEV. T. P. KIEKMAN ON THE THEOEY OF THE POLTEDEA. 161 
janal and contrajanal zoneless poles. Let these be 
fi' being the number of amphiedral and amphigonal together, and q that of the ampbi- 
grammic. 
Every zoneless polyarchaxine has one double zoneless pole, secondary or tertiary; 
every bomozone polyedron has one; every 2r-ple monarchaxine has two (XX.), and 
every 2-ple monaxine contrajanal has one ; and these, by what precedes, are all enume- 
rated. The zonoid signatm'es show in all cases how many of these secondary and 
tertiary or sole axes are amphigrammic. Let the number be 
wZi of amphigonal and amphiedral together ; 
m 2 of amphigrammic poles. 
All the jM; poles, except poles of zoneless triaxines; for they can be nothing 
else, these being the only janal polyedra not already disposed of. 
As each triaxine (XX. ) has three of these poles, then are 
l(fj,-7n,-m2)='L (10, XXXI.) 
zoneless triaxines, on which are q — m 2 amphigrammic axes. Thus the entire number of 
polar edges on these L solids is known^ and consequently the number of their different non- 
polar edges. 
AYe have thus demonstrated that we can enumerate by the data of art. XXXVI. all 
the janal polar P-edra Q-acra and Q-edra P-acra. 
B, C. x-zoned monaxine heteroids (2. XXXI.). 
XLII. Let S be the number of different poles, in Table C (XXXIV.), terminating an 
r-zoned axis of given character and of given zonal signature {Z} or {ZZ"}. Let b' be the 
number of these I poles found on polyarchaxine monarchaxine and triaxine, heterozone 
or homozone, polyedra (XXXII.), in the same signature. All the rest must be heteroid 
poles of which every monaxine heteroid has two. Hence 
i(^-§')=B or =C (XXXL), 
according as r is even or odd, is the number of r-zoned monaxine heteroids having 
this signature. 
J. I'ple zoneless monaxine heteroids (8, XXXI.). 
XLIII. Let £ be the number of zoneless poles terminating on r-ple axis of given 
character Y, in the Table C (XXXIV.). Let be the number of poles terminating 
r-ple zoneless axes of the character Y in the polyarchaxine monarchaxine triaxine and 
monaxine contrajanal, zoneless polyedra. This z, is known by what precedes, and £^=0 
of course, if r> 3, and if the axis be gonoedral, edrogrammic, or gonogrammic. All the 
rest of the g poles are on r-ple monaxine heteroids, which contain each two of them. 
Hence 
and J is given for all values of r>l, and for every character of axis. 
ilDCCCLXII. Y 
