242 PROFESSOK C. J. JOLY ON QUATEPtNIONS AND PROJECTIVE GEOMETRY. 
But there is a wider sense in which these four scalars are invariants, 
are the fourth invariants of any two functions /j and /o, the relation 
If 72 1 and 72o 
({/l//j-(/i/j)«. (/l#2 
(/■./.) ^ (fJA -Lfdf, - tfj,) d) 
= [ahcd) {n — n't + n"d — n’"d‘ + d) (112) 
is evidently true or may be verified at once by repeated application of (16). Thus 
any relation implying a peculiarity of the function / and depending on its four 
invariants, implies also a corresponding peculiarity in the mutual relations of the 
functions and /^/o, that is, of any two functions Fj and Fo decomposible in the 
manner indicated. In particular, if in (112) /o is replaced by f{-^, it is evident that 
the invariants of f\ff\ ^ are identical with those of f. And, moreover, the functions 
may be replaced by their conjugates without altering the invariants. 
We now propose to examine the meaning of a few invariants, bearing in mind the 
remarks of this article, and remembering also that the invariants are more general 
than those of quadrics, for the function /is not supposed to be self-conjugate. 
24. For brevity, replacing/a by a', we have 
n'’'{ahcd) = {a'hcd){ah'cd){cibc'd){ahcd') . . . . (113). 
Ifn'" vanishes, it is 2 ^ossihle to determine an infinite numher of tetrahedra a, h, c, d, 
so that the corners of a derived tetrahedron shall lie on the faces of the origincd. 
For taking any three points a, h, c, and their deriveds a', //, c', three idanes are 
found 
{a'hcd) = 0 , {cd)'cd) = 0 , {ahc'd) = 0 .(114), 
whose common point d enjoys the property of having its derived in the plane 
of a, h, anrl c if, and only if, n'" = 0 . 
Conversely, if this is true for any tetrahedron and its derived, the invariant n'" 
vanishes, and the property is true for mi infinite numlier of tetraliedra. 
Interchanging tlie words corner and face, we have the corresponding inteiqiretation 
of the vanishing of n'. 
More generally, wlien n'" vanishes, an infinite numlier of tetrahedra exists, so that 
the pairs derived from them by the operations of the functions fijfl and //, are 
related in the manner described. 
Analogous extensions will be understood in the sequel. 
25. Again, suppose that the sum of the squares of the roots of n/ = 0 is zero, or that 
n'"'^ — 2n" = 0 .(115). 
In this case, tetrahedra may be found related to their correspondents in such a 
manner that the deriveds of these correspondents have their corners on the faces of 
the originals. 
