PROFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 243 
Of greater interest, however, is the case in which the sum of the square roots of 
the roots of tit — 0 is zero, or when 
— in")^ — 64n.(116). 
Here the n"' invariant of one of the square roots of tlie function (compare Art. 36) 
vanishes, so that by the operation of this square root it is possible, from a suitably 
selected tetrahedron (one of an infinite number), to derive a second, and from that 
again a third, so that the second has its corners on the faces of the first, while its 
faces contain the corners of the third. But directly by the operation ofy’( = /“•/“) 
the third tetrahedron is transformed from the first, and these are so related that it 
is possible to inscribe to the first a tetrahedron circumscribed to the third. 
Similarly, we can interpret invariants arising from relations such as 
++ +.(117), 
where m is the ratio of two integers, and where Q, and are the latent 
roots of f. 
26. Before passing on to invariants of a rather different type, we shall consider the 
relation connecting two quadric surfaces when an infinite number of tetrahedra can 
be inscribed to one and circumscribed to another. 
Let the equations of the quadrics be 
S^^F^^ = 0, Sf/Fo^/ = 0.( 11 ^) > 
let the tetrahedron (ahcd) be inscribed to the first, and let its faces toucli the second 
at the points V, c', cV ; let the function f derive the tetrad of points of contact 
from the corresponding vertices. Then there are four equations of inscription to the 
fii'st quadric 
S«Fu<^ = 0, ShFih = 0, ScFjC = 0, SdF^cZ =0 . . . (119); 
twelve equations of conjugation of the points a', h, &c., to the second quadiic 
Sa'Foh = SIj'Fm = 0 or S«/'Fj 6 = SaFj/^ = 0 . . . ( 120 ); 
and four equations of contact such as 
Sa'F/d =0 or Saf'Fofa = 0 . 
The equations of conjugation require the function Fo/ to be self-conjugate, so that 
.( 121 ), 
and the conditions of contact may therefore be re 23 laced by four equations such as 
SaFo/^ =0 .( 122 ). 
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