240 TEOFESSOE C. J. JOLY ON QITATEENIONS AND PEOJECTRT. GEOMETEY. 
A third root is zero if 
m = Se (1 — or if Se (i — m"^~^ -f e = //;Se<E>~ 3 e = 0 . ( 104 ), 
and this simply requires cl)“'e to he parallel to a generator of the cone, and 
perpendicidar to tlie vector to its vertex. This generator tonclres tlie spliere. 
1 lie condition that the fonrtli root may vanish reduces to 
mT$-A=0 .( 105 ), 
and requires m = 0 for a real function, and in this case the cone breaks into a pair 
of planes, and the symbolic cjuartic degrades. 
xidmitting that T(p~”e — 0 (for an imaginary function), it appears that the 
generator —<p A -f* is common to the quadric and the sphere when four 
roots are zero. 
1 he preceding analysis establishes the fact that a real self-conjugate function may 
belong to the classes, Ij, I„, I3, but not to Ij,. 
A real self-conjugate function cannot belong to if its two united points are 
real, for certain of tlie conditions of self-conjugation of the tetrahedron in the 
limit require Sad = Sa/i — SA = 0, or the line a, h must be a generator of the 
sphere ; and matters are not changed when we assume a and h to he conjugate 
imaginaries. We conclude therefore that no self-conjugate function belongs to I^. 
Since self-conjugate functions of the type 11^ exist, a fortiori they will exist for 
the less restricted types IIj, IIo, II3. 
Self-conjugate functions may belong to the types ITI^, III.i. and to type IV, the 
lines being now conjugate with respect to the sphere (compare the following Article). 
21. If a function converts any tetrahedron into its reciprocal, it is self-conjiigate. 
Here if 
fa = X [bedf fh = y [acd], fc — z [ahdf fd = iv [ahe] . . ( 106 ), 
tlie function producing the transformation is 
fq {abed) — x \bcdi] [qbed) — y [aedf {qaed) + z [a6c/] (qabd) 
— IV [cd)ci] (qabc) . . . ( 107 ), 
which is manifestly self-conjugate. 
Ihis includes as a particular case the deduction from Art. 8. 
Tlie following theorems may be stated here .■— 
If a function has a scalar for a principal solution, its conjugate has three vector 
principal solutions. 
If a function has a line or a plane locus of united points, it has a vector or 
a linear system of vector principal solutions. 
The nature of the function f, which is the negative of its conjugate, has been 
sufficiently considered in Art. 12. 
