242 
MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
Ex. 2. To integrate the equation x 2 dd*+ x d~x ~f(y) ^a=0. 
Putting x=z e , we have 
V 2 u-f(y)^ J = 0 (18.) 
The equation D 2 w=0 gives u—a-\- B0, which we shall substitute in (18.), regarding 
A and B as functions of 6 and y. This gives 
d 2 
D 2 A + 0D 2 B + 2DB — f{y)-y^(z 26 k + 0s 2 *B) = 0, 
whence by the rule 
D2 A+ 2DB-/(3/)^^A=0, 
d 2 
V 2 B-f(y) w t™B=0, 
applied to which, the fundamental theorem gives 
A =2a m i m ^, B =.tb m z m \ 
d 2 
m 2 a m -\-2m b m —f(y)-^a m - 2 =0, 
d 2 
rn 2 b m -f{y)-^b m - 2 =0 ; 
whence writing x for i 6 , and determining a m , b m ; observing also that a 0 , b 0 will be 
arbitrary functions of y , we have 
u =h(^/)+<PM x + ( p2 (y)^ 2 + &c - 
+logx(4 0 (y)+4 l (y)x+4 2 (y)a?+ &.C.), 
%{y)i <Pi (y) being arbitrary, and the succeeding forms of <p m {y ), 4 m(y ) determined 
by the equations 
d 2 
4m(y) —J{y)~y^ 4 m- 2 {y)- 
Euler has exhibited in a series the integral of the equation 
cPu t a ( da ( du\ ( b n 
dxdy'x + y\dx'dy/~4(x + y) 2U ’ 
and on that result are founded many of the solutions of partial differential equations 
in Dr. Peacock’s c Examples.’ We proceed to consider a somewhat more general 
equation, of which we shall give two distinct solutions. 
Ex. 3. Given -^^(x+y^-^f^x+y^+f^x+y^O, 
/i,/ 2 ,/ 3 denoting any functions whatever, to find u. 
Put x=s, x -\-y—t, then dy—dt’ anc ^ transforming, we have 
