24G PEOFESSOE C. J. JOEY ON QTJATEENIONS AND PEOJECTIYE GEOMETEY. 
If the line «, h intersects the connectors in the points and q'j, and if ol, h' inter¬ 
sects them in Qo, (compare (131), (132)), we may write 
« = Qi + hd'i; ^ = Q] + ^oQ'i ; c' = sQi + s'tgQ'i ; d' — sq^ + ; 
a' = sq, -f s't^q'o ; V = + s'Uq\^ ; c = q.t^q '.; d = q, -f t^q ',; 
and the anharmonics of the ranges ahc'd' and a'h'cd are 
(ah){c'd') _ ss'jt^ — G)(fg — ^4) . {a'l/){cd) _ — t ^) 
{he') {d'a) — s't^){s't^ — st^) ’ {b'c){da') ~~ {s't, — st^){st_^ — s\)' 
For a pair of quadrics (118) a quadrilateral on one determines a self-conjugate 
tetrahedron with respect to the other if n and u" of the function vanish. 
Moreover, in this case the quadrics 
S^/Fpj' = 0, S^FgFj^ ^Fof/ = 0 
intersect in a common quadrilateral. 
29. It may be worth while drawing attention to a sinqjle rule for obtaining in a 
convenient form certain scalar invariants of linear functions. These invariants are 
the coefficients of powers and products of x^, x^, &c., in the latent quartic of the 
function 
^1/1 + ^3/3 + • • • + ^11/71. 
and the rule is to distinguish hy accents or suffixes the symbols in {abed) just as if 
this exj^ression had been differentiated. For instance, there is the twelve-term 
invariant 
n^z{abcd) = '^{a-Jj^cd) 
where stands for /pq and a„ for f^a. 
It would appear that when a twelve-term invariant vanishes, every term will 
vanish provided the tetrahedron {abed) is suitably inscribed to a definite curve. 
Suppose eleven terms vanish. Let three be solved for a, and substitution in the 
remaining eight leaves eight equations in 6, c and d. From three of these find b, 
and five are left in c and d ; and on elimination of c, two equations in d remain, 
which represent a definite curve. From symmetry the remaining three vertices 
trace out a curve or curves. These curves are covariant with the functions. 
