229 
EXAMPLES. 
(1.) Let the system proposed for solution be 
(Aptlz + Alby + Ailrs) + (Gru + A303) = Mee” | 
(Vz + AVoig + Ves) — (Uz + Astor) = Bx J 
The reduct equation is in this case 
(dy + ae? + age?)P.u, + (ae? + az). U, = M (dy + Ane? + dye!) a* 
— (M8 + 48°) 8", 
which, as is readily seen, may be written in the shape 
F (2?) . u, = Aa® — Be’, 
where F is a biquadratic of given form, in which the coefficient of the 
highest term has been reduced to unity. 
The solution of this equation is, omitting the arbitrary portion, 
and the arbitrary portion of the solution, itself, is 
Ch, + 0) (—hy)? + Coke? + C2 (- he)? 
+ 
Cshs* + OC", (—ks)* + Osh + O's (- hy), 
if the roots of the biquadratic F (4)= 0 be supposed to be 
ky’, ki, hk, kg. 
The value of w, being thus found, the value of v, ishad by simple sub- 
stitution in either of the given equations, and the mode of determination 
of the arbitrary constants may be easily deduced from the previous ar- 
ticles. 
(2.) If the system of equations proposed for solution were 
(apt, + UA?u, + mAtuU,) + (Av, + a,A°v,) = mat 
(4Vz + O,A70, + MA'V,) — (a, Au, + a:A°U,) = 08" |’ 
we should have for the reduct equation 
‘i F(A’) u, = A’a® — B's", 
the solution of which is, omitting as before the arbitrary portion, 
At B 
“* B@- i)” Fi@-1y} °° 
BR. I. A. PROC. —YOL. VO. : 21 
