MR. W. SPOTTISWOODE ON THE CONTACT OF CONICS WITH SURFACES. 307 
Substituting these values, the condition above written becomes 
A( vh x -wg l -\-Jca l ) 
+ B( wf\—uh x -\-kb x ) 
+ C( ug x -vf ; \l:cy ) 
+D( —ua x —vb x —wc 1 )= 0 , 
and the ratios A : 13 : C : D will be expressed by the determinants of the matrix 
vh 1 —wg 1 -\ : ka x , 
Wfy-Uhy-^kby, 
ugi-vfi+kc,, 
— u(i x — vby — wc , 
X 
y 
z 
t 
1 
n 
K 
A 
But if by analogy with (11) we write 
A„ 
c 1 x—a 1 z—g 1 t= B,, 
a l y—b l x—h l t=C ,, 
/i^+M+ 7i U=D„ 
it will be found that, putting 
f u- \-7j v-\-%w +3- k= U„ 
Ajl + B 1 ??+C,^4-D 1 ^=E 1 , 
the ratios A : B : C : D are equal to 
A,U, — E,m: B, U,— E,v: 0,11! — E,w: D,U, — E,w; 
and consequently when these are substituted in H the terms having E, for a coefficient 
will vanish, and the equation of the locus resulting will be 
(a,..)(A l ,B 1 ,C„D I ) > =0.. ........ (48) 
To find the locus of points at which one of the principal tangents has a four-pointic 
contact with the surface, we must add to the equation H=0 the following, viz. DH— 0, 
which, as has been shown in a former part of this paper, may be replaced by any one of 
the group 
A , 
B , 
c , 
D 
-0. . . . 
. . . (49) 
u , 
V 
w , 
k 
^H, 
But, remembering that p, g, r, s represent the differential coefficients of H with respect 
to x, y, z, t, on the supposition that A, B, C, D are constant, and writing 
(a,..)(A, B, C, D)(a, 13, y , a)=H', 
it is easy to deduce the following system : 
2A 
