38 
MR. HARGREAVE ON THE SOLUTION OF 
If, for example, we apply this theorem to the solution of 
D%+(^*i<2^+^i)DM+(ao'^+&o)M=X ; 
we have the rational fraction 
b\t + io . ZijCt + ^ o ^ + ip 
^^ + Oi^ + «o’ «— (3 ’ /3 — « 
7/ = (DH«iD+«o)“‘(D-«nD-^n^-‘{(D-«)-^(D-/3)-®X}). 
The performance of the operations requires that A, B, &c. shall be whole numbers ; 
and these are the conditions under which the equations are soluble in finite terms. 
If we combine with the above general form the formula (1.), we obtain the solution 
of the equation 
( 4- K) xX-YK-^-\-n{ a^x + D" 
u 
OC ™ 
+ -f &„-3 + (w. - 2 ) ( + V 2 ) ^ - 1 K- 4- V 1 ^ K^r + 
4-...=X 
n—2, 
•^x 
n — 1 n — 2 
'ifx 
in the form 
The important limitation, that A, B, &c. must be whole numbers in order that the 
operations may be practicable, must not be overlooked ; and with reference to this 
point, the attention of the reader is called to the solutions by means of definite inte- 
grals given in a subsequent part of this paper. 
If two or more of the roots a, (3, &c. are equal, we obtain amongst the operations 
expressions of the form g (D-«r, which do not appear to be interpretable in finite terms ; 
but the corresponding solution in the form of a definite integral will apply. 
III. Solution of Equations hy interchange of Symbols alone. 
It has been already observed, that the operation or set of operations denoted by 
any function of D is not of itself intelligible, unless the function is capable of expan- 
sion in integer powers of D, so that fractional operations may not be introduced ; 
but if, by means of the transformation above indicated, the function of D becomes in 
result changed into a function of x, such a result is intelligible, and may be relied 
on as true, although the expressions introduced during the process may be purely 
symbolical and incapable of interpretation. 
Thus — 
regarded as the solution of 
x{T>'^—c^)u-\-2m'Du=yi, 
