144 
Proceedings of the Royal Society of Edinburgh. [Sess. 
and stated his surmise at the meetings of the two Associations mentioned 
above. The immediate result was that Professor J. C. Adams formally 
established the anticipated identity. Taking a set of equations whose 
eliminant is the determinant in question, for example 
«x — kcl -}- Xb -f- [jlc + vd + £ 6 y 
A.3? = Kb + Xc -j- fJid + V6 + £ Ct , 
fxx — kc -+■ Xd + fj.€ 4- vet + £b , ' 
vx — Kd + Xe + /xol + vb + £c , 
= Ke + Xa + fxb + vc + £d ,> 
Adams used on them the multipliers 1 , 6 , 0 2 , 0 3 , 6 4 (0 being an imaginary 
fifth root of 1), thus obtaining by addition 
x(k + 0X + + 0\ + 0±£) = (a + Ob + 0 2 c + OH + fo)( K + 0~'X + 0~y + 0~ 3 v + 
Similarly he obtained 
x (k + o~'x + o~y + e-*v + o ~^) = (a + e~ l b + o~ 2 c + o~h + o-*e)( K + ox+ oy + o\ + o^\ 
and then from the two by multiplication 
z 2 = (a + Ob + 0 2 c + OH + 0±e)(a + 0~ x b + 0~ 2 c + 0~ 3 d + 0~ 4 e). 
This last being free of k , X , i* , v , £ it followed that 
x 2 -(a + 0b + 0 2 c + 0H + 0*e) 
• (a + 0~ x b + 0~ 2 c 4- 0~ 3 d + 0 ~ 4 e) 
was a factor of the determinant with which lie started, and therefore that 
the said determinant was equal to 
— 1 x — ( ct + b 4" c 4" d + b) j - 
• {x 2 - (a ■ 3 rb0 + cO 2 + dO z 4- eO 4 ) ) 
. ( a + bO^ + c0 3 + dO 2 + eO) f 
• {a* 2 - (a 4- bO 2 + c0 4 + d0 + eO s ) ) 
. (a 4- bO 3 + cO + dO 4 + eO 2 ) J . 
Glaisher himself properly points out that since 0 is of the form 
cos |m 7 r + i sin §m7r, Adams’ quadratic factor must be of the form 
x 2 - (A + Bt)(A - Bi) i.e. x 2 -(A 2 + B 2 ), 
a result which is in accordance with the fact of the reality of the roots of 
Lagrange’s determinantal equation. 
Glaisher ’s theorem, it should also be noted, is another instance of the 
assertion of the equivalence of two different forms of an eliminant. 
