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1. Let it be proposed to solve the system of equations in finite diffe- 
rences, of the first order, exhibiting » unknown functions, 
Ung = AU + Oz t+ OWet. .- 
Vay = Aq, + OVe + Cet... 
Wir= Agllg + bVz+ Cet. + - 
&e. 
the auxiliary known functions fi, f2, fs, &c., being, for the present, 
omitted. 
Multiply the first equation by a, the second by «, the third by », 
&c.; then, adding all together, we get 
(Q,A + Mope + Agv+ . . -) Uz 
+ 
ba +b, nee 
ePz (Alig t Unt Wet. ..)= (b,a + nee ye 
(ea + Oye + Csy+ 4 .) Wz 
+ &e. 
Now, as we have introduced » arbitrary coustants, we are at liberty to 
subject them to x conditions, which we may suppose to be— 
A+ Axje+asyt+ ... =kX, 
BAt det byt... =hy,- 
CA + Cy +ey+... =k, 
&e., 
i being a new constant. 
The preceding equation is thus reduced to the form 
ePz (AUz+ Mz tv, +...) =h(Au,+ wrz,+ 0, +...), 
the solution of which is, at once, 
Miz + Vz + Wet... = Ch, 
where C is any arbitrary constant. 
Now, with regard to the quantity £, it is to be observed that if ( — 1) 
of the quantities a, ~, », &c., be eliminated between the assumed equa- 
tions of connexion, the n” quantity will of course disappear of itself, 
and we obtain an equation of the n™ degree in &, and the known quan- 
tities a,, b,, 4, &c., namely, the determinant, 
a, —k, Cae az; , &. 
b, ’ b, -—k, bs ? &e. 
Gas CG, , ¢;—k, &e. at 
&e., &e. 
