224 
where m,, m, are the roots of the equation 
(m—a,-1) (m-6,- 1) =byap. 
(2.) Again, if it be required to solve the system of three simulta- 
neous equations 
A. Uz = b0,+ CW, ) 
A Vz = gl, + Cy y 
A. W,= Agu, + 6,0, J 
we have for the result sought 
“Uns = Cm? + Om" + Cyn" 7 
v, = Dm, + Dyme+D,m,7 >, 
w,= Eym,* + Lym, + H,m,* J 
where ™,, m2, m, are the roots of the cubic equation obtained by the 
elimination of a, «, v, between the equations 
Ange + dv = (m—1)a ) 
br +b» =(m-1)m p> 
CA + Cye=(m—1)v J 
or, 
(m — 1)? — (bya2 + 6,45 + €2b5) (1 — 1) — (¢2b,4, + ¢,a2b,) = 0. 
It will be observed that the form in which, in the correlative example, 
D,, £,, &c., are expressed in terms of C,, C,, C,, remains unaffected, as 
it should. 
4. It is plain that we may employ a method similar to that just 
given, for the solution of the system of simultaneous equations in finite 
differences of the n” order, 
Ursin = ll + be + Ca++» 
Vain = Ag, + bet CoW2t oes 
Wain= Agiz + OVe+ CoWe+... 
&e. 
The reduct equation is, in this case, 
OD (Alig + Vy + VW, +. =H (Ae + Vet Wat. -), 
the equations of condition being 
A+ Apt av+...=ha 
BA + bye t+ byt. . =k" 
COA + Ce +eyt...=khy f’ 
&e. 
