LINEAR DIFFERENTIAL EQUATIONS. 
4.) 
and that of 
oi- 
ls u 
j:(A 2— 2wAw =Q 
Q. 
x’ 
2m 
'x~ X 
A-u ——Am — c^u 
^ X 
2m m m r / 2m m m \ *] 
= {A^ — c2 )''*E‘-‘^"(A — c)~c+i(A+c)«^i'[jc~'(^E~i-«"(A — f)^(A4-c)’'^‘Q^j| . 
The results thus obtained are not interpretable unless B, Aj, A 2 , &c. are integer; 
in the event of any of them being fractional, however, the solution of the equation, 
(the second side being suppressed,) may be found as before in the form of a definite 
integral, by introducing the factor and changing D into 2 , and writing for - the 
integration with regard to ^ between the proper limits. 
Thus it will be found that the solution of 
•• 4- b{) , -f- (ff o'i’ 4 b^u^=0 
between proper limits. 
For taking this value of u^, we have 
nbgZ 
— aY'(f — jS)"^... = {f — «)-''(£* — |S)^... 
Of these two terms, the first vanishes if one limit be z=— oc, and the other have 
the successive values log a, log|8, &c., and the second term is equal to 
{if -ccf^ (s^ - + . . + Zlj , + b^u ^) , 
so that the equation is verified. 
The complete solution therefore is 
/ log a. d„ 
a 4 • • 4 4 «o) ~ ~ 
4&’C. 
or more simply, 
^ {a^v +..+aiM+Mo) 'y“o(y— a)^‘(i; — |3)\...z;' ^dv 
\''^nV''+-’ + 0\V-\-aQ)~'v^o(v-aY^{v—^)^....v^~'du 
+ &C. &c. 
MDCCCXLVIII. H 
