383] 
PROBLEMS AND SOLUTIONS. 
589 
and eliminating the eight quantities, we have the rationalised equation expressed in the 
form, determinant (of order 8) = 0; viz. this is 
A 2 , 
A , 
c 2 , 
A , 
E 0 , 
Fo , 
G 0 , 
0 
B 2 r 2 , 
A 2 , 
G 0 r 2 , 
F 0 r 2 , 
0 , 
A , 
c. 2 , 
Eo 
C 2 s 2 , 
G 0 s 2 , 
a. 2 , 
E 0 s 2 , 
A , 
o , 
A , 
Fo 
A* 2 , 
Pot 2 , 
Eot 2 , 
a 2 , 
C, , 
B, , 
o , 
Go 
E 0 s 2 t 2 , 
0 , 
D,t 2 , 
C 2 s 2 , 
A 2 , 
G 0 s 2 , 
Fot 2 , 
A 
F 0 t 2 r 2 , 
R.,t~ , 
0 , 
B 2 r 2 , 
G or 2 , 
A 2 , 
Eot 2 , 
c 2 
G 0 r 2 s 2 , 
C 2 s 2 , 
B. 2 r 2 , 
0 , 
For 2 , 
E 0 s 2 , 
a 2 , 
A 
0 , 
E 0 sH 2 , 
F 0 t 2 r 2 , 
GoPs 2 , 
B-,r 2 , 
C 2 s 2 , 
D,t 2 , 
a 2 
This seems to be of the degree 18 in the coordinates, but it is probable that the real 
degree is lower. 
[Vol. in. pp. 63—65.] 
1652. (Proposed by W. K. Clifford.)—Through the angles A, B, G of a plane 
triangle three straight lines Aa, Bb, Cc are drawn. A straight line AR meets Cc in 
R; RB meets Aa in P; PC meets Bb in Q; QA meets Cc in r; and so on. Prove 
that, after going twice round the triangle in this way, we always come back to the 
same point. 
Show that the theorem is its own reciprocal. Find the analogous properties of a 
skew quadrilateral in space, and of a polygon of n sides in a plane. 
Solution by Professor Cayley. 
1. The theorem may be thus stated: Given three lines x, y, z, and in these lines 
respectively the points A, B, C; then there exist an infinity of hexagons, such that 
the pairs of opposite angles lie in the lines x, y, z, respectively, and that the pairs 
of opposite sides pass through the points A, B, C, respectively.