38] 
253 
38. 
NOTE ON A GEOMETRICAL THEOREM CONTAINED IN A 
PAPER BY SIR W. THOMSON. 
[From the Cambridge and Dublin Mathematical Journal, vol. i. (1846), pp. 207, 208.] 
It is easily shown that if three confocal surfaces of the second order pass through 
a point P, then the square of the distance of this point from the origin is equal to the 
sum of the squares of three of the axes, no two of which are parallel or belong to the 
same surface (the squares of one or two of the axes of the hyperboloids being considered 
negative); i. e. if 
X 2 
y 2 
z 2 
a 2 + h 
1 
b 2 + h 
"T 
c 2 + li 
X 2 
y 2 
z 2 
a 2 + k 
~T 
b 2 + k 
"t" 
c 2 + k 
X 2 
+ 
y 2 
1 
z 2 
a 2 + l 
b 2 + l 
1 
c 2 + l 
then a? + y 2 + z 2 = a? 4- b 2 + c 2 + h + k +1. 
In fact these equations give 
(q 8 + A) {a 2 + k)(a 2 + l) 
(a 2 — ¥) (a 2 — c 2 ) ’ 
2 (b 2 + h)(b 2 + k)(b 2 + l) 
V ~ (b 2 —£L 2 ) (b 2 - C 2 ) ’ 
2 _ (c 2 + h) (c 2 + k) (c 2 +1) 
(c 2 — a 2 ) (c 2 — b 2 ) 
and adding these and reducing, we have the relation in question; which is also imme 
diately obtained by forming the cubic whose roots are h, k, l.