of Edinburgh , Session 1885 - 86 . 
837 
and suppose the sides when produced to meet the co-ordinate axes 
OX and OY in A., A 2 , A 3 and B., B. 2 , B 3 respectively. Then 
and 
9 A OB B — ^i c 2 ~ ^2 c i) 2 
1 2 W 2 -«A) > 
2 A CA A = ( a i c 2 ~ a 2 c i) 2 
1 2 
with like expressions for the triangles whose 
vertices are at A and B and sides intercepted 
by the axes. 
But, from the geometry of the figure, 
ABC = AB 2 B 3 - BB.Bg + CB.B 2 
ABC = AA 2 A 3 - BA.Ag + C A. A 2 . 
Hence, denoting either of the above ex- 
pressions by (12), we have 
2 A ABC = (23) - (13) + (12) , 
^1^2% l 2 
I I a i^3 il a \b<2, 
by (I.) 
Again, let PQBS be a tetrahedron, the equations to whose planes 
are of the form ax + hy + cz + d = 0, which we may call 1, 2, 3, 4 ; and 
suppose the planes 1, 2, 3, meeting in S, when produced to inter- 
cept on the co-ordinate plane of YZ the triangle A, B, C, 
Then, by the foregoing, 
2AA 1 B 1 C 1 =- 
VA P 
^2 C 3 I ^1 C 3 I I ^1 C 2 I 
Multiplying this expression by the x of the point S, namely, 
I ^l C 2^3 I I <* 1 ^ 2 % I 5 we ^bus get 
I h c A I 3 
6 vol. SA.B.C. = 
b-2 C 3 II ^1 C 3 
b l c 2 || a x b 2 c 3 
In like manner, if A 2 B 2 C 2 and A 3 B 3 C 3 be the triangles intercepted 
