186 
that upon making cc = oo and y-y in the equation to the 
sphere, we must have 
% =z, P = p, Q = f h R - r + 2 (S - s) m + {T- t) m 2 = 0. 
Now the equation to the sphere gives, by differentiation, 
(.V-a)'+(j/'-/3) ! +(*'- 7 ) 3 = 3S 
a’ - a + P — y) = 0, y - j3 + Q (%' — 7 ) = 0, 
l +P 2 +J2(is'- , y)=0, PQ+N(^ / -7)=0, i + Q 2 + 7 t (^- 7 )=o; 
hence, there result between the constants «, ¡3, 7, S, and 
the co-ordinates æ, y, the following relations 
(* - «) 2 + {y - /S) 2 + (nr - 7 ) 2 = 3 2 , 
<2? - a + p 0 - 7 ) = 0, y - (3 + q (* - 7 ) = 0, 
1 + p 2 
% — 
+ r + 2 
7 
The latter gives 
O 
# — 
7 
+ s m + 
1 + q 8 
z — y 
7 = “ 
1 + p 2 + 2pqm + (l + (/°) m s 
+ £ nr = 0. 
0); 
r + 2sm + 
and then a, /3, $ are known from the equations 
oc — a= —p (#— 7 ), y-(3=-q(%- 7 ), ¿ 3 =(«- 7 ) 2 (l+p 2 +9 s ); 
\/l + p 2 + g 3 11 + m 2 + (p + qm) 2 } 
the radius = 
r + 2s?w + ¿m- 2 
Hence we have determined the radius and co-ordinates 
of the center of the circle of curvature of the normal 
section at a point xyz, whose intersection with the tangent 
plane at that point is projected into a line represented by 
the equation 
y — y — m (co — x). 
Cor. Suppose the tangent line to make angles X, ¿t, v 
with the axes of x, p, #, and let cr be the length of the 
arc of the normal section, then (Art. 28 and 131),