LINEAR DIFFERENTIAL EQUATIONS. 
39 
is not interpretable in finite terms when m is fractional. Yet 
[jfi — c2)“’"X}, 
regarded as the solution of 
D{(<r2 — c^)u] — ‘Imxu—Y^, or {x^ — c^)Jiu — %{m—\)xu=^, 
(these forms being derived from the others by changing D into x and x into — D), is 
interpretable for all values of m, and is correct. 
Again, it has already appeared that the solution of 
which is not interpretable in finite terms when m is fractional. 
But if this equation be multiplied by and transformed as before, we get 
{x^-\-c^)Tfiu—2{n—'2)x'Du-\-{{n—\){n—2)—m{m— 1))m=D2Pz=R suppose ; 
and we infer the solution to be 
^ _ D»-« { (^2 _j_ cS)”*- ^D- * { + c2) -’"D”-’”- 'R } } ; 
which is interpretable though m be fractional, provided m—n be an integer ; or the 
equation 
(d;2 c2)D2m — 2aa?DM + 6(2a — 6 -h 1 ) M = R 
is soluble when & is a whole number ; the solution being 
of which a remarkable case is, (a=0) 
~ (suppose) 
M = D"^*'’''^{(<r2 + c2)“*D“'{ (a?2-p.c2)*“‘D*((x2-f c2)X) } } . 
In applying these forms great caution must be used with reference to the introduc- 
tion of constants. The processes indicated in the value of u show &-j-2 constants ; 
which renders it necessary to determine h of them in terms of the other two by 
reference to the original equation. 
Thus, suppose we require the integrals of the equations 
{\-\-x‘^)'D‘^u~2u=x, 
and — 2u—a. 
The form of both solutions is the same, viz. 
