PROFESSOR C. j. JOLY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 207 
where u, v and tv are certain scalars. Operating on this by/’/, we have l)y Art. G 
” (u) ~ ^ ^ il') — i']!} f = 0 . . (457), 
remembering (Art. 103) tliat I" {[>), !'{[>) and /(/>) are the invariants of _/)/. 
But this relation gives Fp {1) linearly in terms of /, 11/(1), (^/(I), and therefore, as 
asserted, the fourth plane will also pass through the common line. 
Hence it appears that (45G), or its equivalent 
[I, Il/il), G;{i)] = 0 .(458), 
represents a double curve on the sextic (455) ; for if^> is any point on this curve, not 
only will (455) be satisfied, but the equation of the tangent jdane at that point will 
also vanish, since every set of three quaternions included in the brackets of (455) is 
then linearly connected. The order of this curve is 7, by Art. 64. 
Moreover, (45G) expresses that a united line of the function f/ passes through the 
point I, or, reciprocally, that a united line of the function/), lies in the plane S/p = 0. 
117. 1 he point determining the /'^'■nction for irhich the plane is a imited plane 
IS a triple j^oint on the sextic. 
If is this point, and if Q, Q, G are the roots of the function f(pgj) answering 
to the united points in the plane, it follows from the fundamental properties of the 
auxiliary functions that 
(/) = NQ . /, Cp„ (/) NAG . (/) = tdd, . / 
(459) 
and consequently the tangent plane and the polar quadric of the point p)^ to tlie 
surface (455) vanish identically. Hie j^oint is therefore a triple point. 
118. It may be noticed that in terms of «, h, c, any three jioints in the plane, 
the triple point is 
Bo = [/(/«)^ ./’'(/M, file)] .(4G0); 
also in terms of these tliree points, if I = \ahc\, 
n;{l) = S [/(pa), h, c,], G/il) = S\a,f(j>h), f(p-)l 
(0 = Uirf^fi piffi pc)] .(461). 
Consequently if q = xa f f -e, we may rejilace the system of equations 
(454) by 
a-A' + y Y -f = 0, xX, + y 3", + = b, -rAb + y lb + = (). (4G2), 
where 
A' = Sa///(/) = (a.Jipa), h, c) ; 
(1) = (ei, J (pa), t (ph), c) fi- {a, f( pa), h, f(pc)) ; 
AY = S«/4' (/) = (a, /( jKt), fipb), f(pc)) . (463); 
and 1, 3 0 , 3^3 and Z, Z.,, Zg may be written down from .symmetrv. 
VOL. CCL—A. 2 Q 
