228 
1 A 
AU, + Wr +0, + &e.= ~;(4 Zt + te. ). F(x) + Ch. 
The operations indicated by the symbols A, A’, &c., being performed, 
the complete solution, in its primary type, is obtained, it being observed 
that a, ~, », &c., enter linearly in the right-hand member. 
A corresponding method of solution, of course, will apply to such a 
system of equations as 
Ur =U, + bv, + ew,+...+fi(a) 
Vz = Alig +b, + Wet... + fy (a) 
Wa= Ag; + OV, + Cr+... 4+ fy © ; 
&e. 
or we may, in some cases with advantage, employ an extension of the 
method stated in the second article. 
8. If the system of equations proposed for solution were of the 
form 
®(A).u,+ ¥(A).02 =F (z) 
@(A).0, —¥(A).u,= F,(2))’ 
where F, and F, are given functions of z, we may proceed in the follow- 
ing manner. 
Operating upon the first equation with (A), and making substitu- 
tion from the second equation, we get 
@(4)?. wu, + ¥ (A). u,= ©(A). F(a) - ¥ (A). F(z). 
The operations susceptible of execution being performed, this equation 
is obviously reducible to the form 
{®(4)?.+ ¥ (4? .}u,= F(x), 
in which there is now but a single unknown function. 
This last equation, in general, admits of solution, and the value of 
u, being found, that of v, is obtained by substitution in either of the 
given equations. 
A mode of solution precisely similar will apply to the system corre- 
lative to the above, namely, 
(aoux + @itz41 + Atzie2+ ... + GnUz1n) + (bovz + byvzr1 + borg +... + bmnvzim) = F,(@) 
(aovz + @y0x:1 + A2Urig tt. +. + QnVzin) = (Boux + byura + boteie +... 4+ Dmttxem) = F,(#) 
or 
® (e?) .u,+ ¥ (e?).v, =F,(2) 
® (6?) .v,— ¥ (e?).u,=F,(«))’ 
the equations being written in their condensed symbolic form. 
