EEV. T. P. KIEKMAN ON ATJTOPOLAE POLTEDEA. 
201 
of (e, are O'G and 0'(G4-1), which are O'E and O' (E— 1), wherefore G=e and 
e is nodal. In the first case, g or (G — 1)(G — 2) is between two 4-laterals and is conva- 
nescible; in the second, g or G(G-f-l) is for a like reason convanescible, so that its sum- 
mits, (^+1) and the r-ace o, will reduce s to a (r-}-2)-edron with a (r+l)*ace, i. e. to a 
pyramid. Consequently s, having the triangle {e, e+1,/]) when e is nodal, and EO' is 
an edge of s, is to be rejected when r>3, because of the vanescible pair 0(E+1) and 
o(6+l). And this is the only s to be rejected in our enumeration for any value of r 
because of such a pair. Adding to this the one above rejected for another reason, for 
r=4, we have to deduct from the ■|{r,(r— 2)-j-2,._,} (r+Sj-edra (S), for every 
value of r, on the account of being generable from the (r-l-2)-edral pyramid. This 
makes the number of (S) thus far ascertained to be 
XXII. It is necessary that we inquire how many of these are repetitions of each 
other, or of those enumerated in the class (Q). 
The leading system of every S is the two edges o?} and oo' in the triangle o'ori ; ori being 
the diagonal in the face F, from the summit ;?=F,. or F^, as the case may be. This on 
is evanescible, and by evanishing, makes oo' convanescible, so that the union of its sum- 
mits gives the pyramidal summit a. Of course the gamics of on' and oo' (OH^ and OO') 
are the first convanescible, and the second, thereby made evanescible. 
We are then first to examine whether any of these (S) have a second triangle a/3y, 
containing a leading system like that in oo'n, whereby S can be reduced to a pyramid, on 
some r-gonal base different from Q ; and then to decide whether this S has been twice 
enumerated above. 
Let ef^ be the generator OO', being the point y+^. Let the summit F, (or F,) be 
n ; then the triangle oo'n contains the evanescible on (or o'n) and the convanescible oo'. 
Let r > 5 for the present : then, if o be not the simplest of the two summits oo', there 
is no pair of summits in S excluding o, whose united rank =:r-}-3. For in general, n 
and e, the only summits not triaces, except o and o', are tessaraces ; and if n is e, it is a 
pentace containing ef^, two sides of the triangle oo'n, and two edges of Q. That is, two 
tessaraces, or a pentace and a triace, are the amplest pair in S, not o' or o; whose 
united rank makes only 8, <r-j-3. 
Now if there be a triangle a/3y containing a leading system di fferent from that in oo'n, 
a. the greatest of its summits must be o, because a+jS gives the same sum with o and o' 
by a convanescing edge. This triangle is therefore 0|3y. And as oy is evanescible, y, 
being neither o' nor n, must be e, the only summit besides these not a triace ; therefore 
this triangle is oe/3 : and /3 being not o', nor n nor e, is a triace ; wherefore, since 
o+^=o-j-o', o' is a triace, and O' is the triangle e(e+l]f,. 
Again, /3 cannot be a summit of O, for oo' is an edge of E, because e is in OO' ; and 
OE is an edge of S, because oe is. If, then, /3 were in O, o/3 could not become conva- 
nescible by the evanescence of oe, because its summits o(3 are in O and E collateral in 
MDCCCLVII. 2 E 
