INVESTIGATION OF THE CURVATURE OF SURFACES. 37 
2°. The condition that the first three of equations (6) should hold simultaneously 
may be written 

d fe d fa d fo 
AE Bee Cs 2—() 18 
das Fie VA dc ay 
where A», B,, C,, are minors of the determinant 
' Ging big OG 
. Ay bs, © | 
as, ds, eg 

The geometrical interpretation of this condition is that the normal to the quadric at 
the extremity of the line whose direction cosines are proportional to a;, by, ¢, (which there- 
fore has à “ a ; =. proportional to its direction cosines) lies in the same plane with 
1 3 - 


the lines - anaes and 2 -—= Ÿ — =. For (18) shows that it is perpendicular 
a L I 3 : 
to the line 
which is perpendicular to both of these, since we have 
a, A,+ bd, B,+¢, OC, =0 
Gage Ge Bie en (a0 
3°. The geometrical interpretation of equations (8) is that 
a, LH D y G4 2=0 
is a tangent plane to the cone 
(a-k) 2° + (6-k) y+ (c-k) +2yz+2mix+2nxy—0 
along the line ~- = 4. = * 
sz b; C; 
3 

Circular Sections. 
We may find the conditions that the section of the quadric be a circle by considering 
it as a conic which has an infinite number of axes. 
Now taking the equations of the axes from (15) and putting them, for brevity, in the 
shape 
a (ie SE 
AM ST Da EC: 
we see that this will happen if the denominators be severally zero. 

Hence, expanding, we get the conditions 
A, = be? + cb? — 2 Ib, c, — k (BP + ¢2) = 0 
B, = cay + acy — 2 me, a, — k (ef + a?) = 0 
C; = ab? + ba? — 2 na, 6, — k (a5 + 67) =0 
whence we get 
be? + cb? —2 Ibe, __ ca? + ac? — 2 me, a, _ aby + ba —2n & b, 
by + ci FF ey + & wi ay + br rik GE) 

