308 
PEOFESSOR C. .1, JOL’i’ ON OUATERNTONS AND PROJECTIVE GEOMETRY. 
Ihe diiect inteipietation ot this identity is that the traiistomiatioii converts the 
Jacohian I{p) = 0 into two surfaces, one beino- the Jacobian J (q) = () and the other 
the surface of the twelfth order 
J {qY J {qY 1 "' { })) -\- J Yi) I" Yp) — 1 ' { p) — h 
(487). 
This surface is the locus of three of the united points of functions wdiich have a 
zero latent root, the fourth united point lying on the Jacobian J Yq) = 0. 
The ciitical curve is triple upon this surface, and the surface meets the Jacobian 
again in a residual curve of tlie eighteeiitli order, wliicli is the locus of united points 
corresp)onding to a doidde zero root. 
128. Making the substitution s = t - J Y/) in the latent quartic of the function 
f{p, q) the equation reduces to 
- 1'"{p)) + 6-3(6 ./YiY — sr"{qj) JYi) + r'{p)) 
+ — 3/'" (p) J{qf + 21” {pj) J(q) — F (p)) = 0 . (488). 
A second root of the original quartic is equal to J Yq) if 
AJYiY ~ {p) '^YlY + 2/"(y>)'/{i/) — I' {p) = h . . . (489), 
and this is the locus of united points winch correspond to double latent roots. This 
surface is of the twelfth order, the critical curve is a triple curve upon it, and it 
meets the Jacobian in the same curves as (487). 
The locus of united points corresponding to triple latent roots is the curve of 
intersection of this surface with the surface of the eighth order 
QJYif - 'iF”{j>)JYj) + 7"(jo) = 0 .(490). 
But the critical curve is doulJe on this surface, and accordingly it counts six times 
in the intersection, so that the locus of triqde united pioints is a curve of order 
36 (= 8 X 12 - 6 X 10). 
129. Further, quadruple united points are the points common to the surfaces (489), 
(490), and 
UYl) - I"'Yp) = ^ .(491), 
which do not lie upon the common critical curve. 
In order to calculate the numljer of these quadruple points it is necessary to find 
the numbei of points common to the critical curve and the curve locus of triple 
points. Now 24 (= 4 X 3 X 2) functions have triple zero roots, this being the 
number of points common to the surfaces lYp) = 0, F i^p) ^ 0, F' Yp) = 0 in the p 
space, and the curve locus of triple points being of the 36*'’' order meets JYl) — h in 
144 points. Subtracting 24, there remain 120 points on the critical curve. 
Ihe triple curve therefore intersects (491) in 24 quadruple united points, and in 
