133 
fought, in which Randolph E. of Murray was slain, and the 
King himself led into captivity. — Believe me, Sir, with 
many thanks for your obliging communication, your faithful 
humble servt., 
Edin., 4 June, 1802. Walter Scott. 
“Note on ‘An Essay on the Resolution of Algebraic 
Equations, by the late Judge Hargreave, LL.D., F.R.S./” by 
the Rev. Thos. P. Kirkman, M.A., F.R.S. 
The following appears to me to be a demonstration, that 
Dr. Hargreave’s very skilful attempt to solve the quintic is, 
like all that have preceded it, a failure : 
One of the final equations at p. 94 is — 
V\ -Vy* — Ai^i — b x )-\-k 2 (z 2 — &i)+A 3 (% — &i)+A 4 ( 2 4 — -&i)+ A 5 (z 5 — b x ) 
=-2(A<). 
If so, it is possible to find a x -\-a x =iA l , a 2 -\-a 2 =A 2 , <fcc., such that 
y^+y 2 T =(( 2 aOO T +(( 2 “ 0 ")= : ( p5 ) ir +(Q 5 )b where 
P 5 — ^and Q 5 —y r 
As y x -\-y 2 —2{51), p. 85, is a rational and symmetrical function of 
the z’s, P 5 +Q 5 is one also; and this is plainly impossible, unless 
P-{-Q— 2A£ is a symmetrical function of the Vs, That is 
3 /i"+ 2 / 2 5 == 2 (A^)=AS^AS( 2 — 6i)=0; and y x — —y 2 ; 
or yx+y s =2(5 1)=0, 
This destroys the form of the conditioned quintic in z, at p 82 ; 
for (51)=0 gives, as (21) and (31) are each =0, (p. 78), (p. 7)> 
(5 l)=46f — 56 1 5 4 -p6 5 =0 ; or b 5 is expressible in b x and & 4 . 
I suspect that it follows in a similar way from the equation in 
p. 54, that (41)=0, in the quartic in z in p. 53. 
It is evident from consideration of the known solutions of the 
quartic, the cubic, and the quadratic, that every solution of the 
general quintic F=0, whose roots are x a x x x 2 x 3 x 4 , must be a 
transformation, by algebraic artifice, of the expression 
5 x 0 — 4^ 0 + ( a + a 2 ■ -f- a 3 + a 4 ) (x x -j- x 2 -f- x 3 -f- # 4 ) ; 
