CEETAIN APPLICATIONS TO THE THEOKT OF DEFINITE INTEGEALS. 
797 
But since from (1.) x has only one value, viz. v, it is evident that '2f{x)=f{v). Hence 
/W = ©[/(^)]^ 
Now lety’^^^M=U represent any differential equation with constant coefficients, 6 being 
the independent, u the dependent variable, and U a given function of To secure the 
utmost generality, I will suppose a rational fraction. Thus if the equation were 
^ we should have 
/(!)= 
Now 
“={/(i)}’‘u=e[/(;*-)-](|-;*^)"'uby(2.)=e[/(*)-].-{Jr-wo+^W}, (3.) 
"^(x) being an arbitrary function of x. This is the complete solution of the differential 
equation proposed. The reader will have no difficulty in applying it to particular cases. 
The arbitrary constants have their origin in the complementary function ’>^(x). 
In the Calculus of Residues the theorem corresponding to (2.) is the following : — 
/(^)=E^^+ T ■■ (4.) 
The first term in this expression fox f{v) is equivalent to the result of the first part of 
the operation 0 in the second member of (2.), viz. that which depends upon the develop- 
ments which are effected in ascending powers of the simple factors of the denominator 
of f{x). The second term is a transformation of the result of the second part of the 
operation 0. It would be more conveniently expressed in the form 
-c.Ai, 
-j^V — X 
The solution of the equation/'^ w=U, furnished by (4.), -will evidently be 
-/-) i 
yp(x) being an arbitrary function of x. 
When, as indeed is usually the case, is ^ rational and integral function of the 
d 
symbol the second member of (5.) reduces to its first term. The symbols 0 and T 
then become identical. 
2ndly. From the equation v — x= 0, we have, by the general theorem of transformation, 
2/(a^)c?a^=0[/(a^)]~ 
