196 
EEV. T. P. KIEOIAIv^ ON AIJTOPOLAE POLTEDEA, 
the evanescent parallel ; and thirdly, every pair drawn from the extremities of that nodal 
parallel which is a side of fl. That is, there are \{r-\-V) of these parallels whose non- 
contiguous pairs are 3)(r— 1), which is the number of pairs ^and e'f, of fellow- 
generators which do not cross each other, and of the (r+3)-edra Q generated by such 
pairs. 
We suppose next that ed is one of the i(r— 3) nodal parallels, and that Me, having a 
fellow-generator liUc', does not meet ee'. Then h!Jd uill not meet ed ; for ed, parallel to 
two sides Mi' and M' of the 4-lateral Jik, h!k', does meet hJJd unless it meets hk. And for 
a reason given in this article the (r-{-3)-edron generated by ed and hk cannot differ from 
that obtained from de and KJd. Every Q obtained by a pair ed, hk, is obtained also 
from de, h!k'. 
Now let ed and^' be two of the -1(^—3) nodal parallels. They do not meet; and 
they generate a (7’+3)-edron Q which cannot be generated by any other pair ed, hh' or 
Ml', ii' ; for, if it could, this other pair would be two fellows of ed and ff, which have 
no fellows. Thus each of the -g-.(r— 3)(r — 5) pairs of nodal parallels will generate a 
distinct (r-j-3)-edron Q. 
From all this it appears that the number 11', of different (Q) obtained from the pairs 
of diagonals that do not cross each other, is (r>3 and =2f?i-j-l) 
n:=i{E.-|-i.(r-3)(r-l)-fi.(r-3)(r-5)} 
= 3)(r— 4)(r+l)-l-3(?"— 3)(r— l)-f-3(r— 3)(r— 5)}. 
XVII. It appears, I say : and it is certain, that none of these Q obtained from opera- 
tions in the base H, are alike in the relation between the faces OO'O" and the other- 
faces of the figure. Any two of them, Q' and Q", present different arrangements of the 
circuit of base summits 1 2 ... r, which enclose those three faces ; so that Q' and Q" can- 
not be reduced to a pyramid having the base O whose summits shall be those signed 
1 2 . . r, by similar operations with vanishing gamic pafrs. This is plain, because Q' and 
Q" have not been constructed by similar operations upon the base O and the vertex oj. 
But it remains to be proved that Q" has not three faces different from OO'O", which 
by proper selection of vanescible gamic pairs may be brought to form an r-gonal base fl". 
If such a thing is possible, this Q" may have the relation to the contoiu- of that Q'', which 
Q' has to that of fl ; and thus the operations that gave us Q' uill have genei-ated the 
same (r-l-3)-edron with those which gave us Q", and vice versd. 
We are then to examine whether Q" has tliree summits, not oo'o", which can by con- 
vanescing edges be united to form an r-gonal summit 
Now Q" has no summits besides oo'o", except 1 2 . . . r. Of the three summits of 
which we are in search, two, one, or none may be among oo'o". The three may be oo'e, 
oed or ede". 
There may be a convanescible edge oe, the edge F.(F-l-l). Of these two faces, F and 
F-j-l, neither can be a pentagon ; for the lines fg, fh, evanescibles making the pentace/’, 
can neither of them he a side of fl. If, then, oe convanesces, F and F-f-1 
