586 
PKOBLEMS AND SOLUTIONS. 
[485 
The lemma is at once proved by means of my theorem for the relation between 
the distances of five points in space, {Cambridge Mathematical Journal, vol. II. (1841), 
p. 269, [1],) viz. if the point 1 is the centre of the circumscribed sphere, and the points 
2, 3, 4, 5 are the points A, B, C, D respectively, then the relation in question, viz. 
becomes 
0 , 
(12) 2 , 
(13)*, 
(14)*, 
(15)*, 
1 
(21) 2 , 
0 , 
(23)*, 
(24) 2 , 
(25)=, 
1 
(31) 2 , 
(32) 2 , 
0 , 
(34)*, 
(35)*, 
1 
(41) 2 , 
(42) 2 , 
(43) 2 , 
0 , 
(45 ) 2 , 
1 
(51) 2 , 
(52) 2 , 
(33)*, 
(54) 2 , 
0 , 
1 
1 , 
1 , 
1 , 
1 , 
1 , 
0 
0, 
r\ 
r*, 
r 2 , 
r 2 , 
1 
r 2 , 
o, 
A*, 
a 2 , 
1 
r~, 
h\ 
o, 
A 
A*, 
1 
r\ 
A> 
o, 
c 2 , 
1 
r 2 , 
a 2 , 
A*, 
c*, 
o, 
1 
1, 
1, 
1, 
1 , 
1 , 
0 
Multiplying the last line by — r 2 and adding it to the first line, this is 
r 2 , 
o, 
o, 
o, 
o, 
1 
r 2 , 
o, 
h 2 , 
g 2 > 
a*, 
1 
r 2 , 
h\ 
0, 
A> 
A*, 
1 
r 2 , 
A 
A, 
o, 
c*, 
1 
r 2 , 
a 2 , 
A*, 
c 2 , 
o, 
1 
1, 
1, 
1, 
1 , 
1 , 
0 
and then proceeding in the same way with the first and last columns the equation is 
— 2r 2 , 
0, 
0, 
0, 
0, 
1 
0 , 
0, 
h 2 , 
«*, 
1 
0 , 
A*, 
0, 
A, 
A*, 
1 
0 , 
/ 2 > 
o, 
c*, 
1 
0 , 
a 2 , 
A*, 
C*, 
0, 
1 
1 , 
1, 
1, 
1, 
1, 
0 
which is in fact the equation of the Lemma. See my papers in the Quarterly 
Journal of Mathematics, vol. ill. (1859), pp. 275—277, [286], and vol. v. (1861), 
pp. 381—384, [297].