244 PROFESSOE C. J. JOEY ON QUATERNIONS AND PROJECTR'E GEOMETRY. 
An infinite number of tetrahedra may consequently be respectively inscribed and 
circumscribed to the quadrics 
SpFo/~(/ = 0, SpFop = 0.(1^3), 
when the condition (121) is satisfied and when the n"' of f vanishes; and if this is 
likewise possible for the given quadrics, we must have 
FJ’2 = F„ or /^ = F,-^F„ or /= (F.-^FJ^^ . . . (124). 
The sum of the square roots of the latent roots of the function F 2 ''^F^ must 
consequently vanish, or the invarianh^^ (US) of this function is zero. 
It has l)een proved incidentally in this article if a tetrahedron circumscribed to 
S^/Fop = 0 is self-conjugate to SpFg^' = 0, that the invariant n"' of the function 
Fj^^Fg is zero ; and if the tetrahedron is self-conjugate to S^Fgp = 0 and inscribed 
to Sr/Fpy = 0, that the same invariant of the function Fg^^F^ is zero. Here 
F3 = Fo/ 
It must be carefully observed that irr dealirrg with quadrics the extent of the 
irrvariance (Art. 23) is lirrrited. If F^ arrd Fg are self-corrjrrgate, the functions yjF^y’o 
and t iFo/o rnrrst be self-corrjrrgate before theorerrrs carr be exterrded fronr the quadrics 
deterirrirred 1ry the simpler to those deterirrirred by the more complex fuuctiorrs. 
27. The irr variant vJ' vanishes if 
{a'Ucd) + {a'hc'd) + {a^hcd') + {aVdd) + {aVcd') + {ahc'd') = 0 . (125). 
To save verbiage irr the interpretatiorr, the edges ah arrd c'd' may be called the 
opposite edges of a tetrahedrorr arrd its derived. If each edge of (abed) irrtersects the 
opposite edge of [a'b'dd'), the irrvariarrt will manifestly varrish, for every term will 
be zero. 
To display the natrrre of the corrditiorrs requisite for deternrirring a tetrahedron 
possessirrg this pr'operty, when n" = 0, let a arrd h be assrrmed fixed, and then five of 
the terms nray be writterr irr forrrrs 
Scfpl =0 { 11 = 1, 2, 3, 4 or 6).(126), 
where is orre of five lirrear qrraterrriorr furretiorrs. Three equatioirs give 
<^ = Udhfzd,_f\d~] .(127), 
and sirbstitrrtiorr irr the forrrth arrd fifth require the point d to be orr the curve of the 
quartic srrrfaces 
(128), 
* This condition iippcars to iinswcr in every particnhir the condition (compare ‘ Elements of 
Quaternions,’ New Ed., vol. ii., p. 377) that a triangle can he inscribed to one conic and circumscribed to 
another (see, however, Salmon’s ‘Three Dimensions,’ Note to Art. 207). 
