225 
and the solution of the reduct equation is, if «, #, «’, &c., be the n se- 
veral roots of unity, 
AU; + MV, +, +...=O(ak)*+ C' (w@k)*+ CO" (aky... 
5. In the same manner, if the system of equations to be solved had 
been 
A” Uz, =AU, +b, + CWrte.. | 
APU. = AU, + bv, + 6,0, + oe* 
A”, Wz= Agliz + Dp + CWrt... [’ 
&e. 
we should have for the reduct equation 
A”. (Alp +e tpt... .)=h (Au, + ev, + w,+...), 
the solution of which is, «, «, «, &c., being, as before, the ” several 
roots of unity, 
AUz + Vet z+. ..=C(ak+1)*+ C0! (#@k+1)*+C"(wk+1)y +... 
EXAMPLES. 
(1.) Let it be proposed to solve the system of the second order 
Urn =O, + OW; “| 
Vinx = AU, + CW, ? e 
Wig=Agllz+ bv, J 
The equations of condition in this case are 
Ane + Ay = Pr ) 
ba + by = pe : 
CA +eye=hy J 
the reduct equation 
2 (Ali, + KV, + 9,) =F (AU, + KV, + ¥W;) ; 
the solution of this equation, 
Az + eV, + vw, = Ch* + 0’ (-k)*; 
‘while the equation to determine & is 
HP — (b,y + C105 + C2b,) k? — (¢2b,a, + ¢,4,b;) = 0. 
(2.) If it be proposed to solve the system 
A?. U, = 60, + CW > 
A? Uz = Az + Cz >, 
Aa. = Astle + bv, J 
