402 
Mr. Babbage’s essay towards 
assume tyy = A+ A (log.y) 2 -p&c.-}- A (log.y) 2 " 
o i n 
£ y becomes A Ay 2 -{- Ay 4 -f- &c. + A y» 
012 n 
x}/ e~ y becomes A -}- Ay 2 -{- Ay 4 &c. + A/* 
012 n 
By comparing the co-efficients, 
A = ~ a A = % a, &c. A — \ a 
o o i i n n 
and calling the right side of the equation F (y 8 ), if 
vP £> + r-- = Fy 4 , 
a particular solution is 
■^y = i'p {(iog->)*} 
and similarly if 
\]s £ y — \Pe— =sF(y) = a function containing only odd powers 
ofy, one particular solution is 
+y = i F {log.jv} 
and the general solutions may be readily deduced as above. 
Problem V. 
To reduce the equation \^x -j- A x % a x &c. -j-Nix 
\p v x + X = o, to one in which the last term is wanting, by 
means of a particular solution, 
Let jx be the given solution, make\p.r =/x -}- <p x, and 
substituting this value, the equation becomes 
J' x — |- Ax -J- B x y.J {3 x &c. -|- N x x J" v x -j- X ^ 
-{•<$> x + Ax y.<pa,x + Bxx(pfix-{- See. -f- N X x a? v X 
the upper line is by hypothesis equal to nothing, therefore 
the equation is reduced to this, 
<p x + A X x <p a X -j- &c. -j- N X X <P V X = o’ (l) 
and if we can discover the general solution of this latter equa- 
tion, that of the former may be readily found. 
