588 
PROBLEMS AND SOLUTIONS. 
[383 
To develope the solution, I notice that the expression for the equation $ (a, b, c,f, g, h) — 0 is 
b 2 c 2 (g 2 + h 2 ) + c 2 a 2 (h 2 +f 2 ) + a 2 b 2 ( f 2 + g 2 ) 
+ g 2 h 2 (b 2 + c 2 ) + h 2 / 2 (c 2 + a 2 ) + f 2 g 2 (a 2 H- b 2 ) 
— a 2 / 2 (a 2 +f 2 ) — b 2 g 2 (b 2 + g 2 ) — c 2 h 2 (c 2 + h 2 ) 
— a 2 g 2 h 2 — b 2 h 2 f 2 — c 2 f 2 g 2 — a 2 b 2 c 2 = 0 ; 
see my paper, “ Note on the value of certain determinants &c.,” Quart. Math. Journ. 
vol. in. (1860), pp. 275—277, [286]. Or, as this may also be written 
2 \{b 2 + c 2 — a 2 ) (g 2 h 2 + a 2 / 2 ) — a 2 / 4 } — a 2 b 2 c 2 = 0, 
where 2 refers to the simultaneous cyclical permutation of (a, b, c) and of (/, g, h). 
Hence we have only in this equation to write 2A — r, 2A — s, 2X — t in place of 
(/, g, h), and to omit the terms independent of A, being in fact those which are equal 
to <j) (a, b, c, r, s, t). Observing that we have 
g 2 h 2 + a 2 / 2 = |4A 2 — 2A (s + t) + st} 2 + a 2 (2A — r) 2 
= 16A 4 — 16A 3 (s +1) + 4A 2 (s 2 +t 2 + 4st + a 2 ) — 4A [si (s + t) + a 2 r] + s 2 t 2 + a 2 r 2 ; 
/ 4 = (2A — r) 4 = 16A 4 — 32A 3 r + 24A 2 r 2 — 8Ar 3 + r 4 , 
the equation becomes 
16A 4 [2 (b 2 + c 2 — a 2 ) — 2a 2 } 
— 16A 3 [2 (b 2 + c 2 — a 2 ) (s +1 ) — 22a 2 r} 
+ 4A 2 [2 (b 2 + c 2 — a 2 ) (s 2 + t 2 + 4st + a 2 ) — 62a 2 r 2 } 
— 4A {2 (6 2 + c 2 — a 2 ) [st (s +1) + a 2 r] — 22a 2 r 3 } = 0, 
where the 2’s refer to the simultaneous cyclical permutation of the (a, b. c) and the 
(r, s, t). The coefficients of A 4 and A 3 are, it is easy to see, each = 0 ; and in the 
coefficient of A 2 the terms 2 (b 2 + c 2 — a 2 ) (s 2 + t 2 ) — 62a 2 r 2 are = — 42a 2 r 2 ; hence dividing 
the whole equation by 4A, we find 
A [2 (b 2 + c 2 — a 2 ) (4si + a 2 ) — 42a 2 r 2 } — [2 (b 2 + c 2 - a 2 ) [si (s + t) + a 2 r] — 22a 2 r 3 } = 0, 
which is the required relation between (r, s, t). 
It may be noticed that, expressing the distances r, s, t in terms of Cartesian or 
trilinear coordinates (x, y) or (x, y, z), then r 2 , s 2 , t 2 are rational and integral functions 
of the coordinates, and the form of the equation therefore is 
A. 2 + B 2 r + C 2 s + D 2 t + L 0 st + F 0 tr + Gr 0 vs = 0, 
where the subscript numbers denote the degrees in regard to the coordinates. Multiply 
ing this equation successively by 1, r, s, t, st, tr, rs, rst, we have eight equations linear 
in the last-mentioned eight quantities, the coefficients being of known degrees respectively ;