306 ME. W. SPOTTISWOODE ON THE CONTACT OE CONICS WITH SURFACES. 
But if H 0 represent the Hessian of U, then BC — $ r2 ='H. 0 (v 1 w l — u 2 ), . . ; and the expres- 
sion in question will he equal to the product of the Hessian into 
w x u x —v' 2 , . . v'w'—UiU 1 , . ,)(a, b, c,f, g, A) 2 , 
where a, b, . . f, . . are the six coordinates of the line about which the cutting plane is 
supposed to revolve, as given in equations (9). The two positions of the cutting plane 
will coincide, if either the Hessian or this expression vanishes ; the latter, regarded as 
a relation between the coordinates of the line, expresses the condition that the plane of 
section may contain both principal tangents, i. e. may be the tangent plane, as may be 
verified by the following considerations. If the plane of section coincide with the tangent 
plane, then 
A : B : C : D=w : v : w : k ; oy . ., 
which, being substituted in the expression in question, will cause it to vanish identically ; 
or we may proceed otherwise, thus : regarding A, B, C, 1) as the constants of the cutting- 
plane, the equation IT = 0 is 
(a, ..)(A, B, C, D) 2 =0; (42) 
the equation of the plane itself may be written 
A|-f B^+C^+D^=0 ; (43) 
and since this plane passes through the point (x, y, z , t), we have also 
Ax-\-~E>y-\-Cz-\-T)t=0 \ (44) 
by means of which three equations the ratios A : B : C : D may be determined. By a 
process quoted by Professor Cayley (Quarterly Journal of Mathematics, vol. vii. p. 1), 
the solution of these equations depends upon the square root of the quantity 
• • ( 45 ) 
a, 
t , 
, 
X, 
5 
T> 
— H 0 W], 
w'. 
l ' , 
X, 
1 
% 
23 , 
f , 
iB, 
y-> 
w'. 
»i , 
u' . 
ml, 
y> 
Jf , 
c, 
Z, 
s 
v' , 
u ' , 
ri , 
Z, 
iB. 
#1, 
D, 
t, 
& 
l 1 , 
m', 
n ' , 
h, 
t. 
a 
y > 
z , 
t . 
X , 
y » 
2 > 
t , 
• 
. 
l , 
n , 
K > 
a . 
. 
? , 
1 1 , 
K , 
. 
. 
which is in fact identical with the expression (41). 
The locus of the points for which one of the principal tangents meets a given line, 
say, the line (a y , b x , c u /„ g„ 7q), will be found by eliminating A, B, C, D from the 
equations (42), (43), (44), combined with the following condition: 
a /i+ J /i + c/ h+«i/+%+c 1 A=0 . (46) 
But since the principal tangent is the intersection of the plane (A, B, C, D) with the 
tangent plane, we have 
h h c,f, g, A, = 
A, B, 
c, 
D 
U , V , 
w, 
k 
(47) 
