THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 175 
5. When n=5, assume 
X=H=X + G &y+ Az Ly + Ay Xy~+ A, 2X5, 
Xg=X1 + Ay Lat Ay X34 A, U4 Ay 2s, 
Xs=Ay + dy Xy+ Ay Wzy+ Ay Uy+ My Ws, 
Xy=X, + Ay e+ As Ly + Ay Ly+Q, Ws, 
and 1 4(#) 3X7 Kg KX, Ky. 
Then the condition necessary and sufficient for the sym- 
metry of the terms in 2* is 
p l=a, G7 a3 a; 
The corresponding conditions for the terms in 2}, and 
vv are 
Ya=F ay Ay A,= Aj Az Az + A, 2 y+ A, G2 y+ Ay Az a 
= Aj My Ag+ Gy Ay AZ 4+, As A} + a3 Ay Ay 
= Aj As Ay + A, Wy As + A, Ay Wj + Ay A My 5 
and those for terms in #2 x are 
BA, Uy = Ay My Ag+ My A Ug + Ay G3 A+ Uy 3 A+ Gy G+ Gy a3 
= 2a, dy ds a,+ af a2 + a} a3 + a3 a} + a2 a. 
Tt will not be necessary to exhibit the rest since the above 
are sufficient for the determination of a. The first and 
second indicate that a, a, a, and a, are the roots of an 
equation of the form 
a'— Aad’+ Ba?- Aa+1=0, 
and, combining them with the others above given, we find 
Sa Sa, dy A3= Ja? Ay dg +40, dy Az Ay; 
or A’?=38A+4; and A=4, or -1. 
Again : — i 
Yd, Mg = Ay Ay Us Oy + 4(3a+ Fay az) 
(= B(2G, )? + 2a — BA Ay As — 2A) Ay As A; 
or B=tB?-$A-2. 
Hence the three systems 
A=4, B=6, or, =4;,A=-1, B=I, 
give rise respectively to the relations 
(a—1)*=0, 
(a-1)*-—10a=0, 
and a+e@+a+a+1=0. 
