PEOFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY 245 
which is complementary to the sextic curve (compare Art. 64), 
(129). 
Selectiug any point d on this complementary curve of the tenth order, c is 
determined by (127), and the sixtli condition must be satisfied. 
Hence it appears that any two vertics may be assumed at random, and a plane 
locus for the third. Ten points d lie in this plane, and ten tetrahedra satisfy the 
conditions. 
Generally, also, if the sum of the j^roducts of the square roots of the latent roots 
of the function vanishes, an infinite number of tetrahedra may be found related to 
their correspondents, so that corresponding edges a, h ; a', 6', are intersected by 
opposite edges of intermediate tetrahedra. (Compare Art. 25.) 
28. The case in which the two invariants n' and n'" vanish simultaneously is of con¬ 
siderable importance in the theory of the linear function. These conditions are 
always satisfied for the functions 2/^ —f~ f \ and also for functions of a more 
general ty]3e; in fact, for functions whose squares satisfy a depressed equation 
U'J + ri'T- + = 0, or (/^ - ,s^^) {P - 6-'^) = 0 
(1.30). 
It appears from Art. 24 that two systems of tetrahedra exist, one set having their 
correspondents inscribed to them, the other set being inscribed to their corre¬ 
spondents. We shall prove that one system of tetrahedra exists ivhich are at once 
inscribed and circumscribed to their correspondents. 
Let (h and be the united points of f for the roots i s, and q\ and for the 
roots dt si Take any line whatever 
q = xipiif'IUP 2 )vf. 2 ) (x*, ?/variable) . . . . (131), 
intersecting the lines qyq^ and q\q'^. The function f converts this line into the 
line 
p = xs{q^ — uq^) -f yd{q\ — vq'„} .(132), 
which intersects the connectors of the united points in the harmonic conjugates ol 
the points of intersection of the original line. Repeating the operation, the line p) is 
restored to q. 
In other words, when n' and n" vanish, the transformation interchanges lines ivJuch 
cut harmonically the connectors of the united points ; or it transforms a certain con- 
genency of lines into itself 
Take any tetrahedron having opposite edges, ab and cd, on two conjugate lines of 
this congenency ; the corresponding tetrahedron has the two edges c'd' and a'U 
respectively on those two lines, and either tetrahedron may be said to be at one and 
the same time inscribed and circumscribed to the other. 
