'[160 
to be 
>ponding 
C" are 
which 
160] ON A THEOREM RELATING TO RECIPROCAL TRIANGLES. 7 
and expressing the other minors in a similar form, the equation to be proved is 
(GHA + HFA' + FGA") (By - C/3) ^ 
+ (GHB + HFB' + FGB") (Ca - Ay) ! - 0, 
+ (GHC + HFC + FGC") (AA - Ba) J 
HF 
A’, 
B', C 
+ FG 
A", 
B", 
C" 
A, 
B, C 
A , 
B 
c 
a , 
A, 7 
a , 
A , 
7 
The first determinant is 
~{a(BC' -B'C) +/3(CA' -CA) + y (AB' - A'B)} = - ^ (««" + /3(3" +yy”) = - ^G, 
and the second determinant is 
{a (B"C-BC") + A (C"A - CA") + y (A"B - AB")} = ^ (aa' + AA' + W) = 
and we have therefore identically 
HF(- G) + FG(H) = 0. 
The corresponding theorem in geometry of three dimensions is that a tetrahedron 
and its reciprocal have to each other a certain relation, viz. the four lines joining the 
corresponding angles are generating lines of a hyperboloid, or, what is the same thing, 
the four lines of intersection of corresponding faces are generating lines of a hyperboloid. 
The demonstration would show how the theorem in determinants is to be generalised. 
2, Stone Buildings, Lincolns Inn, February, 1855.