ME. W. SPOTTISWOODE ON THE CONTACT OE SUEFACES. 
281 
expressions in Y, Z, T, &c., we shall obtain the following values for the ratios sought, 
viz. — 
But 
-l 
(Y, Z, T) (Z, X, T) (X, Y, T) (X, Y, Z) 
( 04 ) 
k(Y, Z,T)=-m(X, Y, Z), 
7c (Z , X, T) = -fl(X, Y, Z), 
A'(X, Y, T)=-w(X, Y, Z). 
Hence, omitting the common factor (X, Y, Z), we have 
a h S 1 
u v w k ’ 
(05) 
and proceeding in a similar manner with the equations in h, b, f, m ; g, f, c, n ; 1, m, 
n, d, it will be found that, when the quantity (X, Y, Z) does not vanish, 
V=(a, b, . ., f, . .1, . .)(x, y, z, t) 2 ={ux J rvy- J i -wz+ict ) 2 ; .... (GO) 
that is to say, that the only quadric which in general has a superficial four-pointic 
contact with U at any given point P is the tangent plane taken twice. 
From the relations given above, it appears that if any one of the equations 
(Y, Z, T)=0, (Z,X,T) = 0, (X, Y, T)=0, (X, Y, Z)=0 . . (07) 
is satisfied, then all are satisfied ; so that it will be sufficient to study any one of them. 
Although I have not succeeded in reducing the expression (X, Y, Z) to any very simple 
form, it may still be worth while to show how it may be extricated from the condition 
of a compound determinant. With a view to this, we may write down the values of 
X m , . . in full, viz. : — 
Xm = 
3 , 
7 5 
Y ul = 
7 , a , 
3 5 
v , 
w , 
k. 
W , U , 
^115 
®115 
Al5 115 
^115 
-^- 112 ”^ 
3 , 
7 5 
a , 
+ 
3' 5 
7' 5 
Y„ 2=2 
7 , a , 
i , 
+ 
7 ' 5 
cd , 
V , 
w , 
A 5 
V , 
w , 
k , 
w , 26 , 
& , 
W , 
26 , 
7-’ , 
?!» 
^125 
^125 
^115 
®115 
**12, jy 125 
^125 
Ai, 
Pll, 
S 115 
X 123 =2 
3' j 
7 r 5 
V, 
1 
3 , 
7 5 
^ 5 
Y 122 =2 
7 5 ^ 5 
y , 
+ 
7 5 
a , 
& , 
V , 
w , 
Jc , 
'« 5 
W , 
li , 
20 , u , 
k, 
ID , 
U , 
/r , 
5l2> 
^125 
®12J 
$225 
^ 22? 
^225 
^ 125 yy 125 
«125 
^22? 
P 225 
•S 2 25 
— Vooj — 
P 9 
7 r 5 
Y — 
x 222 — 
7 ' 5 a' 5 
y, 
v , 
W , 
k, 
w , u , 
2225 
^225 
S 22l 
^225 P225 
^ 22 ? 
All” • ’j T n , — 
! • -5 - 1-112 
Z 122= • T 1SS — 
Zqoo - 
T — 
X ooo 
• 5 x 222 
