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NOTICES OF COMMUNICATIONS TO THE 
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moreover they lie in a hyperboloid. The quadric surface, instead of being defined as 
above, may, it is clear, be defined by the equivalent conditions of touching each of the 
8 given lines: that is, we have the envelope of a quadric surface touching each of 
8 given lines ; these lines not being arbitrary lines, but being a system of a very special 
form. By what precedes, the envelope is a quartic surface. It appears, however, that 
in virtue of the relation AF + BG + CH = 0, this is no longer a proper quartic surface, 
but that it resolves itself into the above-mentioned hyperboloid taken twice. That 
is, restoring the original A, B, &c., in place of A 2 , B-, &c., the envelope of the 
quadric ax 2 + by 2 + cz 2 + dw 2 = 0, where a, b, c, d vary, subject to the condition 
Abe + Bca + Cab + Fad + Gbd + Hcd = 0, (which is in general a tetrahedroid), is when 
A, B, C, F, G, H are the squared coordinates of a line (or, what is the same thing, 
when \/AF + YBG + \/CH = 0) a hyperboloid taken twice, viz., this is the hyperboloid 
passing through the given line and through the symmetrically situate seven other lines.} 
November' 12, 1868. pp. 103, 104. 
Professor Cayley gave an account to the Meeting of a Memoir by Herr Listing, 
“Census räumlicher Complexe oder Yerallgemeinerung des Euler’schen Satzes von den 
Polyedern,” published in the Göttingen Transactions for 1862. The fundamental theorem 
is a relation a — (b — tc) + (c — k + nr) — (d — k" + tt' — co) = 0 existing in any figure whatever 
between a the number of points, b the number of lines, c the number of areas, 
d the number of spaces, and certain supplementary quantities k, k, k" , nr, nr', co. In 
an extensive class of figures these last are each = 0, and the relation is a — b + c — d = 0; 
thus, in a closed box, a = 8, b = 12, c = 6, d= 2 (viz., there is the finite space inside, 
and the infinite space outside, the box): if the box be opened, a = 10, b = 15, c = 6, 
d = 1; if the lid be taken away, a = 8, b = 12, c=5, d = 1; in each case, a — b + c — d = 0. 
If the bottom be also taken away, a =8, 6=12, c = 4, d = 1; but here one of the 
supplementary quantities comes in, k." = 1, and the theorem is a — b+c — (d— /c") = 0. 
The chief difficulty and interest of the Memoir consist in the determination of the 
supplementary quantities k, k! , k", 7r, it', co. 
December 10, 1868. pp. 123—125. Appended to Paper by Mr T. Cotterilf “On a 
Correspondence of Points etc.” 
Observations by Professor Cayley and Mr W. K. Clifford on the connexion of the 
transformation with Cremona’s general theory, and the analytical formulae. 
According to Cremona’s general theory,—taking {x u y x , zi) and (x. 2 , y 2 , z 2 ) as 
current coordinates in the two planes respectively,—if we take in the first plane, any 
three points 1, 2, 3, and any other three points 4', o', 6', then if X 1 = 0, F x = 0, Z x = 0 
are quartic curves, each having the double points 1, 2, 3, and the simple points 
4', o, 6', we have a transformation x. 2 : y 2 : z. 2 = X x : Y x : Z x leading to a converse 
system 
Xi : yi : z 2 = X, : Y, : Z, 
of the like form ; viz., there will be in the second plane three points 4, 5, 6, and 
three other points P, 2', 3', such that A" 2 = 0, F 2 = 0, Z., = 0, are quartics having the 
double points 4, 5, 6, and the simple points F, 2', 3'.