1888 .] Prof. Anglin on Theorems mainly Alternants. 383 
Hence, if (12) denote any one of the above three expressions for 
areas, we have 
area ABC = (23) - (13) + (12) 
Aa&c I l^n^^ I ^ 
a h c 
a h c 
a h c 
• • • 
Zg ??^g %g 
Zj Tfn^ 
Z3 
Z3 TWg 
Zg mg Wg 
Again, employing Quadriplanar co-ordinates, and taking ABCD as 
the tetrahedron of reference, let PQRS be any tetrahedron the 
equations to whose planes are 
Zjtt + -i- n^y + = 0 , (l,m^n,r)2 = 0 , 
(l,m,n,r)^ = 0 , {l,7n,7i,r)^ = 0 , 
which we may call 1, 2, 3, 4 respectively ; and suppose the planes 
1, 2, 3 meeting in S, when produced, to intercept on the co-ordinate 
plane BCD(a = 0), the triangle B'C'D'. Then it may be shown that 
area B'C'D' = 
ABCD I m^n^r^ \ ^ 
BCD 
BCD 
BCD 
m 2 ?^2 rg 
m^ % ^’i 
77%^ 7\ 
mg 7^g rg 
w^3 r^g ?*3 
?7^g ?^g rg 
j 
where A, B, C, D denote the areas of the faces of the tetrahedron 
of reference. 
To find the value of the a co-ordinate of S, which is the point of 
intersection of the planes (?, m, r\ = 0, (Z, m, r)^ = 0, (Z, m, w, r)g = 0, 
— solving these equations we have 
g ^ ^ y 
I i - I ^1^2’ 3 I 1 
3F 
ABCD 
Zj 7n-^ n-^ 
Zg Wg Wg rg 
h “3 »3 H 
I 1 
where V is the volume of the tetrahedron ABCD. 
Hence, multiplying the above expression for area of B'C'D' by 
the value of a furnished by this equation, we get 
VOL. XV. , 1 / 11/88 2 B 
