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MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
this value of u involving but one arbitrary constant, the coefficient of the first term 
only of its development will be arbitrary, and the rest will be formed in subjection to 
the law proposed. 
Ex. 2. Let the equation be Mm-ry^qzT) am + h Um 
By the theorem we have 
f(m) a l m + b i f(m) a. 2 m + b 2 
■«*— ' +M=2) is+y^-2=FW- 
u 
/(D) «,D+/, 
/(D) « 2 D + 6 2 
' /(D — 1) aD + b V(D-2) «Dt6 z m=0 1 ^ n ) s -r c oi“ c i 5 • 
Assume 
, «iD + ^ , a 2 D + b 2 9 . 
f-f- L. : f Vf-f- .-r> . i S 2 V— V. 
«D + 6 
«D + 6 
n „ /(D) „ /(D) 
Here P, tAL-At = P.,4 ; 
trary, 
i/(D-L)- , 2/(D. 
C Q 
whence observing that are still arbi- 
u—f{ D)v, 
v + 
z,D + b 
«D + b 
i . . a 2 T) + b 2 m=co F(m) . 
Vv 4- - S fsm J e 2# M = 2__„ -zt7x-s b, '+ c 0 -f c/. 
a 
D + 6 
* m =o /(m) 
From the second of these equations, which is linear, and of the first order, the 
complete value of v will be found, whence that of u will be obtained by differentia- 
tion, or by the solution of a differential equation with constant coefficients, according 
as the form of f(D) may determine. 
D. § 3. On the Theory of Generating Functions as connected with Equations of 
Partial Differences. 
We shall confine our observations on this subject to the case of equations involving 
two independent variables, the most general form of such equations being 
%{mri)u mM +p 1 (wm)M M _i.„ + p 2 (»m)M«- 2 .» . . . -j 
+^ 0 (?im)M m .„_i+^ 1 (»m)« m _i.»_i 4 - , ^ 2 (»iwjtt w _ 2 .»-i . . . j =f(mn). 
J ryA mn ) u m.n~2 +% 1 (»m)w m _i. w _2 +X 2 ( wm ) M «-2.»- 2 ... J 
The above equation may be placed under the form 
2 <p (m fum^r.n-r 1 (69.) 
the forms of <p(nm), and the value of r and r 1 , being different in different terms of 
the equation, the greatest difference of the values of r we shall represent by i, and 
the greatest difference of the values of r 1 by i 1 . 
Let u be the generating function of u mn , its development ~2{u mn x m y n ) being arranged 
in ascending positive powers of the variables, the lowest index of each being 0. By 
reasoning similar to that employed in the preceding chapter, it may be shown that 
the equation for determining the value of u will be 
2{<p(D,D’,)s r6 +r’6' u } = 'Z[ffnn)z m6Jrn6 '} 
+ (!///) + Cl h(^V • . + O i _i(6')6 ( *- 1) ' 
. . . (xx.) 
