800 PROFESSOK C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY'. 
twenty iuterssctions, but I propose to show that these in reality correspond to ten 
contacts. 
Take, for example, the curve (476), and let be a point of intersection and take 
j>^ to be the Jacobian correspondent of (/., so that/’(/>!, = 0 . Then the tangent to 
the curve at 7 , is 
[/(Ui'/)> /(/V/i)> /Ovu)] = 0.(478). 
But this tangent lies in the tangent planes at the same point to the .system of 
cpiartics (./'(Bi 7 )>/(iVf)’/(B iT) + ^' 7 ) = where u is arbitrary, and as these 
quartlcs contain the critical curve, the sextics touch this curve where they meet it. 
122 . Hence, t/te locus of the Jacohiaii correspondents o f points on the ciutical curve 
is a curve of the tenth degree ; for in the plane [pip-:pf] there are ten points which 
are Jacobian correspondents of points on the critical curve. 
The Jacohian quartic of tJie plane \_P\qrp)-f\ contains ten lines. 
The tangent plane to the Jacolnan cpiartic at a point on the critical curve, corre¬ 
sponding to one of the ten points just mentioned, intersects the plane of Art. 120 in 
a line wJiich transforms into a plane cubic on the sextlc surface into which the 
tangent plane to the*quartlc transforms. But the quartic transforms into a tangent 
plane to this sextic, and therefore contains the cubic, consequently the quartic 
contains the line. 
12 - 3 . We shall now consider the orders of the surfaces and curves into which given 
surflices and curves are transformed hy the relation connecting and q (434). 
With an arbitrary surface Q = 0 in the q space is associated a complementary 
Q' =: 0, so that the points of the two surfaces compo.se tetrads of united points of 
functions of the four-system. These two surfaces, of orders ni and in' respectively, 
transform into a common surface of order n. 
An arl)ltrary line in the p space cuts the surface (a) in n points, and to these 
corre.spond 4/i points in the q .space situated on a sextic curve (475). This curve cuts 
the surface Q in 6ni points, and the.se are generally united points of 6ni distinct 
functions, hecause the surface Q arbitrary. Hence n = Gin. 
Again, the sextic cuts the surface Q' in 6 /?J points, but the.se tall into triads of 
united points complementary to the Gni points. Hence n = ^ Gad; and we have the 
complete formula 
n = Gni = '2ni' . 
iMore generally, if the surface Q is wholl}^ composed of sets of v united points, 
6ni Gin' 
There is a ca.se of exception fur a Jacobian quartic {q, /{jGq), J / (iVi)) = ^ 
which transforms into a plane and not a. surface of the sixth degree as (480) vould 
gi\e for I — ni = 4. But here the sextic curve cuts the quartic in 4 points and 
