223 
Writing these equations in the form 
(dy + @”) Uy + Oy, + CY, +. = 
0) 
All, + (by — €?) Vy + Cyldz +... =0 
Agu, + 6,0, + (6, -€”) Wz +. sar 
&e. 
and remembering that the symbol e” is commutative with constants, we 
get as the result for w,, the determinant 
fae, b,, G, &e. ) 
Ag, b,—e?, C2) &e. . U, =, 
a,, 6, ¢s~e?, &e. 
&e., &e. J 
the first term of which is, of course, 
(0-6) (B,— 6) (05-0). 0. te 
Consequently, the result is an equation of the form 
Uren + LU ina PUzin2 + &e. + ou, = 0 ) 
where «, 8, y, &c., c, are constants; and if the roots of the correspondent 
symbolic equation be, as before, /,, k, h,, &e., the value of wu, is, as 
stated, 
U, = Ck," 3 Che? 35 Ck + &e. 
3. If the system of equations to be solved had been 
A . Uz = Alex + bz + Wz +++ ) 
A. Ug = Ait +b, + 6,00, +... | 
A. W,= Alt, + OV + Cgbzt... f’ 
&e. 
it is evident that the solution required will be had by simply substitut- 
ing in the results of the previous article (a +1), (b.+1), (¢+1), &e., 
for @,, b2, ¢,, &c., respectively. 
EXAMPLES. 
(1.) Thus, if it be required to solve the system of two equations 
A. Ug = Uz + by, 
A.V x = Ally, + Oe } j 
we haye, at once, 
U, = Cym,* + Cxm,", 
m,—a,-—1 Mz—,—1 
v= o,( "| m+ OC, a) m,", 
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