206 
EEV. T. P. KTRKMA^y' OX ATJTOPOLAE POLTEDEA. 
united rank of whose summits equals 5+4, the united rank of o and T/ in the triangle 
oo'tj. Therefore a=o also when r=5, and all the reasoning from article XX. is 
applicable to that value of r. 
AYhen r=4, we see from 
14238241 
that erj is not an edge, unless either 
H=2 and e=l, 
or 
H=4 and e=3, 
two equivalent conditions, either of which gives the heptaedron (12J of art. XX., which 
is to be rejected on account of its being reducible to the 6-edral pyramid. The only 
other system, H=2 = e, and H=4=e, both alike, is the S' having two leading systems, 
made with fellow-generators, m being =1 in the formula r=4ra. No other has two 
leading systems. Therefore all the reasoning of articles XXII — XXVIII. apphes 
equally to every value of r> 3. 
The number of (r+3)-edra (S') having a triangidar O' which we constructed in XIX. 
is exactly r, namely, half the number of generators X, which all pair themselves into 
fellow-generators Of these we have proved that there are always three which 
have only a single leading system ; and it has just been sho'UTi that, when r is even, there 
is always a fourth, havhig two leading but fellow systems. The number to be deducted 
from our enumeration on the score of their being repetitions of some other S', is therefore 
i(^-3-2.) 
(XXI.), to be subtracted from 
i(r.(7— 2)-2+2,_,)~0^^-+ 
The remainder is 
i(r^_3r+2)-o(’-‘‘^'=:n;;, 
the number of (r+3)-edra (S) thus far known, which is to be added to D^+II^ m 
XVIII. 
XXX. It remains that we inquhe how many of these (S) are repetitions of those (Q) 
enumerated in XVI. If Q' one of this class (Q) is one of (S), it has an evanescible edge 
the united rank of whose summits is r+3. The highest rank of 0 0 ’ or 0 " in Q', the 
summits into which the vertex of the pyramid was broken, is r — 2 ; for the united rank 
of the three +r+4, and the smallest have together at least six edges. There must 
therefore be in Q' a pentace distinct from 0 0 ' and 0 " ; for 7’+3 — (r— 2) = 5. Now Q has 
this pentace only when the generators ef and eh are drawn from one summit e. Then 
this evanescible edge, if it passes through one of 0 d 0 ", the greatest of which is 0 , must he 
oe, and 0 ' and 0 " must be triaces. Again, if oe in Q' be the evanescible edge of the leading 
system of S, there will be a convanescible edge through either 0 or e, which vanishing 
after oe, will make 0 or e an r-ace. The only summits not triaces in Q' are 0 , e, h, andf; 
and it is impossible that the summit 0 after the loss of the edge oe, can unite with either 
h or/', which are both-4-aces, to make an r-ace; for r — 3+4<r+2. 
Therefore the convanescible edge of the S-system of Q' must be e(e+ 1 ). But e+ 1 >■ 3 ; 
