2o8 Mr. Babbage on the analogy between the 
tial service in the improvement of this latter branch of ana 
lytical science. One of the first which presents itself is the 
method of solving the differential equation 
o —ydx n -fi A dx n - l dy -f Bdx n — 2 d 2 y -j- &c. + Nd”y + Xdx n 
compared with that of solving the functional equation 
o ~ -tyac A -tyoLX Itya'a? -j- &c. 4 “ "Ntya”— l x -f- X 
X X X 
Euler and D’Alembert succeeded in integrating the dif- 
ferential equation when all the coefficients except X are 
constant quantities, and Lagrange, in the Memoirs of the 
Academy of Berlin, 1772, explained. a method of treating the 
same when they are variable ; both the processes alluded to 
depend in the first instance on reducing the equation to the 
solution of the same equation, wanting the last term ; and 
this is effected by means of a particular integral of the given 
equation. 
The solution of the functional equation is precisely similar, 
it is first reduced to the solution of 
0 = + Aij/aa? -j- B -\10dx -{- &c. -j- 
and this is effected by knowing a particular function which 
satisfies the original equation : this process I have already 
given in a former paper. The resemblance between the 
solutions of the two equations continues also in this respect, 
if foe is a particular case of the given equation, and Ka?, Ka?, Ka?, 
* % 
&c. are particular cases of the same equation without its last 
term, then 
fyx=fx- f-Kx. x(x,ctx,. .tt. n -'x)-\-Kx .x(jc,ax, . . o, n —'x) + &c. 
is a general solution of the equation. 
