PROBLEMS AND SOLUTIONS. 
579 
705] 
The expression of the function F is in effect given in my paper, “Note on the 
value of certain determinants, &c.,” Quarterly Mathematical Journal, t. III. (1860), 
pp. 275—277, [286]; viz. a, h, c being the edges of any face, and /, g, h the remaining 
edges of a tetrahedron, then 
volume = {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 ) -f h‘ 2 / 2 (c 2 + a 2 ) + f 2 g 2 (a 2 + b 2 ) 
— a 2 f 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 }, 
where, when the tetrahedron becomes a quadrilateral, the volume is = 0. 
In this formula, changing c, b, h, g, f, a into a, h, c, d, x, y, we have the required 
equation F = 0; viz. this is found to be 
a 2 b 2 c 2 + b 2 c 2 d 2 + c 2 d 2 a 2 + d 2 a 2 b 2 — b 2 d 2 (b 2 + d 2 ) — a 2 c 2 {a 2 + c 2 ) + x 2 y 2 (a 2 + b 2 + c 2 + d 2 — x 2 — y 2 ) 
+ x 2 (a 2 c 2 + b 2 d 2 — a 2 d 2 — b 2 c 2 ) + y 2 (a 2 c 2 + b 2 d 2 — a 2 b 2 — c 2 d 2 ) = 0, 
which, with xy = ac + bd, determines x, y. Substituting in the foregoing equation for 
xy its value, the equation becomes 
(ad + be) 2 x 2 + (ab + cd) 2 y 2 = 2 [a 2 b 2 c 2 + b 2 c 2 d 2 + c 2 d 2 a 2 + d 2 a 2 b 2 + abed (a 2 + b 2 + c 2 4- d 2 )), 
or 
(ad + be) 2 x 2 + (ab + cd) 2 y 2 = 2 (ad + be) (ab + cd) (ac 4- bd). 
To show more clearly how this equation arises, I observe that we have identically 
F - (a 2 + b 2 + c 2 + d 2 — x 2 — y 2 ) (xy + ac + bd) (xy — ac — bd) — 2 (ad + be) (ab + cd) (xy — ac — bd) 
= {(ad + bc)x — (ab + cd) y} 2 . 
The resulting equation (ad + be) x — (ab + cd) y = 0, together w T ith xy = ac + bd, gives 
for x, y the foregoing values. 
[Yol. xxi., pp. 81, 82.] 
4392. (Proposed by S. Roberts, M.A.)—If N p denotes the number of terms in 
a symmetrical determinant of p rows and columns, show that the successive numbers 
are given by the equation 
N k - iVjfe-j - (k - l) 2 N k _ 2 + i (k -1) (k - 2) [N k _ 3 + (k - 3) = 0, 
k being positive and jY 0 being taken equal to unity. 
Solution by Professor Cayley. 
It is a curious coincidence that the question of determining the number of distinct 
terms in a symmetrical determinant has been recently solved by Captain Allan 
73—2