EQUATION AND ITS TRANSFORMATIONS. 
775 
whence 
Id ddu 
dx 3 a ' u x? dir 
then 
9. Let the above value of u be expanded in powers of h, so that 
tt=C aV <* ,+ *« = P 0 + P 1 A + P 8 fe2 . . . +IVt + P; +1 /t +l d-&C., 
d 2 P ;+1 0 , 
dho 
~—cru = 
Id ddu 
x 2 did 
+ 
a 2 Pi +1 A' +1 + & c., 
dx 3 
and therefore P i+1 satisfies the differential equation 
ddu „ i(i + 1 ) 
Thus the general integral of this differential equation is 
u = A. coefficient of h i+1 in expansion of e a ^ +xK> 
+ B. coefficient of h i+l in expansion of e -a^+x&) 
The particular integrals to which the different modes of expansion of e aV ^ +x ^ lead 
will now be examined, and connected with the forms already obtained in § I. 
10. The coefficient of h i+1 in the expansion of e a ^ +xh) is equal to the coefficient 
of h i+l in 
1 + a(x~ -J- xhb) 1 -f- 9 ( ( x~ + xh) -f- — (cc 2 -fi xh) 1 -f- —,(£c 3 -f- xh)~ R ( (x--\-xh) ,J + &c., 
a? 
"5V 
and the coefficient of h 1+1 in (x z -\-xKf ( ' 2n ~ 1) 
(n-h)(n--l) , , , (n — i — %) 0w _._ g 
(<+!)! ' 
Thus the coefficient of li l+l in the terms involving uneven powers of a 
L / JT 1 T 
2 (i + 1)! 2 ' 2 
. a—l ,~~x 2 r 
3. 5. 
2 ’ 2 
Oi 1—7T 
61 
l a i _Dll 3 
2 (i + l )! 2 ' 2 
/• U -d, 1 a 2 * 2 , 1 
2 ^‘ | 1 
(7 —i)(l-f) 2^2! 
&c. [ =\U, 
