MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
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wherein z e —x, z 6 '—y. The summation 2 in the second member extends from m — 0 
to m=co , and from n= 0 to w=x . The functions <D 0 (/), <I> 0 (g*), &c. are in general, 
but not always, arbitrary and independent. Their forms are in each particular instance 
to be determined by the initial conditions of the problem. 
Ex. 1. Let the proposed equation be u mM — 2u m . n -i — 2ic m -\. n -i = 0. 
This is one of Laplace’s examples. Applying' the theorem, we have 
u — 2z 6 'u— 2z e + =<1> 0 (/)-}- T 0 (s^, (70.) 
+d / 0 (a i ) ("7 1 1 
u 1 — 2y — 2xy u '' 
To show in what way the arbitrary functions are to be determined, let it be supposed 
that when y— 0, u—f(x), a known function, and when x=0, u=f(y), a known func- 
tion also, then in (71.) making successively x=0, y — 0, and x and y together =0, we 
have 
■ — r~-2y — r=My)> ... (72.) 
%(0)-i-W 0 (x)=f(x), (73.) 
^o(0)+^o(0)=/(0), (74.) 
(72.)X(l-2y) + (73.)-(74.) gives %(y) + ^ 0 (x)=f(x) + (l -2y)f 1 (y)-f(0), whence 
A x ) + (!- 2 v)fi (y) -/( Q) . 
1 — 2 y — 2 xy 
U : 
7YI 771 
Ex. 2. Let Um.n ~}“ I ]Um— l.n T" #2 v m.n— 1 “k \ Um — 1.»— 1 = /H j 
Here, by the theorem 
D 
D 
Assume 
whence 
ii-\-a l ^-^i 6 u^a 2 z 6 7i-\-b^-^i 6 + 6 'u = z s + e -f<E>(g* , ')-|-‘'F(g‘’). 
v-\-a l t*v-\-a 2 z 6 'v-\-fa e + ( 'v~'V, then | — p~jTy’ 
u = (jy-\-\)~ l v=i- 6 (D)- l i 6 v=~^Jvdx, also V=(D-j-l)U 
= 1 ) {i x+y + 1 ®(y) + ‘ T(. r )) ■ = . (a? ■ + 1 )g*+y ■ + F(.r ) + F 1 (y ) , 
the functions F(a) and F x (i/) being arbitrary. Hence the equation determining v 
becomes 
v -f- a x xv -f- a 2 yv -}- bxyv = (x + 1 )z x +y + F(a>) + Fj( y), 
(% + l)e x+!/ + F(r) +F t (y) 
1 + a x x + a^ij + bxy 3 
u ='- r d yy)^’+n-njM (75 .) 
xj 1 + a x x + a£) + bxy 
Had the second member of the original equation been 0, we should have had 
u=±- fix , F W+ F -M (76.) 
x*/ 1 + a x x + a^y + bxy 
Suppose it here, as before, required to determine the arbitrary functions by the 
