298 
Froceedings of Royal Society of Eclinhurgh. 
Note on Professor Cayley’s Proof that a Triangle and 
its Reciprocal are in Perspective. By Thomas Muir, 
LL.D. 
(Read April 4, 1892.) 
The vertices of the original triangle being 
(tti , , y^) , {a^ , ^2 5 72) 5 ’ /^3 5 73) 5 
the conic being 
^2 ^2 ^ ^2 _ 0 , 
and Aj , , 
C 
1 5 • 
A = 
. . . being defined by the nine equations 
L%lL, 
I “'1^273 ! 
involved in the single matrical equation 
«1 
/3i 
7i 
-1 
A: 
A. 
^3 
«2 
A 
72 
= 
Bs 
«3 
A 
73 
Cl 
c. 
C 3 
the equations of the lines joining the corresponding angles of the 
two triangles are found to be 
(Bi 7 i - + (Ciai-Aiyi)y + - \a^)z = 0^ 
(^272 ~ ^2^2)^ “t (C 2 ct 2 — Asf^t^y + (A2/32 — '^2^2)^ ~ ^ r 
(^373 ~ d- (Cgu^ - A^y^y + (Ag^3 — Bgag);^ = 0^ ; 
and what is required, of course, is to show that these lines must 
meet in a point ; i.e., that 
^i7i - - ^i7i ^1^1 - 
^272 “ ^ 2(^2 ^ 2^2 ~ -^272 -^ 2(^2 ~ ®2^2 “ ^ * 
^373 ~ ^3^3 ^3®''3 ~ “^373 “^3/^3 ~ ^3®3 
Professor Cayley effects this by a transformation of the elements 
of the determinant, and then by developing at length the deter- 
minant so found.* 
Portunately this tedious process is quite unnecessary, as the sum 
* See Quarterly Journal of Math., i. pp. 7-10, or Collected Math. Papers, 
iii. pp. 5-7. 
