384 
Proceedings of Royal Society of Edinburgh, [june 18 , 
vol. SB'C'D' - 
VABCD 
B G D 
Uo To 
I 
BCD 
m-. n. r. 
BCD 
m-, n-, 7\ 
A B C D 
7\ 
1.2 7)12 ^2 ^^2 
h ™3 % »’3 
In like manner we shall obtain three similar expressions for the 
volumes of the tetrahedra having their vertices at S, and bases in 
the other planes of the tetrahedron ABCD, namely, 
K 1 l^n2r. 
3 
Z 1 Zi7772r3 
3 
AC D 
A C D 
A CD 
J 
A B D 
AB D 
AB D 
I 2 ^*2 ^ 2 
h H n 
Zi 774 r4 
7.2 7 II 2 7*2 
h 
Z 4 m 4 r 4 
^3 ^3 ^"3 
Z 3 77g 7*3 
1.2 772 ^2 
Z 3 m 3 rg 
^3 ^3 ^3 
I 2 7772 ^’2 
and- 
K\lgii2n^\^ 
ABC 
I 2 n%2 ri2 
?3 m 3 ?^3 
ABC 
?3 m 3 «3 
ABC 
Z 9 m^ ??9 
where K~ 
VABCD 
ABCD ^ 
\ m^ rj 
?2 ^2 ^2 ^2 
Z3 m3 W3 r^ 
while corresponding expressions exist for the tetrahedra similarly 
formed, and having their vertices at P, Q, R respectively. Hence, 
observing the geometrical property stated in the former paper, if 
(123) denote any one of the above four expressions obtained for 
volumes, we have 
vol. PQRS = (234) - (134) + (124) - (123). 
VABCD I 
ABCD 
1.2 7??2 ^2 
h ^3 
Z 4 W 4 1 \ 
ABCD 
Zj m^ 7\ 
Z3 7?^3 7*3 
h % *4 
4 i' c z> 
Z 4 m^ 7"i 
Z 2 m 2 7^2 7*2 
Z 4 7774 774 r^ 
ABCD 
Zi 7774 TZj 7*4 
1.2 7772 ^2 ^2 
Zg m3 77g 7 3 
( 2 ). 
2. We will now give proofs, of a uniform character, of the 
theorems (1) and (2), and show how to arrive at a generalisation 
of them. 
Employing a more convenient notation, the theorem (1) may be 
enunciated as follows : — 
If (12) denote any one of the three expressions 
b.c.. 
I «i^2 I 
I «1^2 I 
