In applying the calculus of symbols to partial differential equations, we find an extensive class with coefficients involving the independent variables which may in fact, like differential equations with constant coefficients, be solved by the rules which apply to ordinary algebraical equations; for there are certain functions of the symbols of partial differentiation which combine with certain functions of the independent variables according to the laws of combination of common algebraical quantities. In the first part of this memoir I have investigated the nature of these symbols, and applied them to the solution of partial differential equations. In the second part I have applied the calculus of symbols to the solution of functional equations. For this purpose I have given some cases of symbolical division on a modified type, so that the symbols may embrace a greater range. I have then shown how certain functional equations may be expressed in a symbolical form, and have solved them by methods analogous to those already explained. Since (
x d/dy - y d/dx
) (
x
2
+ y
2
) = 0, we shall have (
x d/dy - y d/dx
) (
x
2
+ y
2
)
μ
= (
x
2
+ y
2
) (
x d/dy - y d/dx
)
u
, or, omitting the subject, (
x d/dy - y d/dx
) (
x
2
+ y
2
) = (
x
2
+ y
2
) (
x d/dy - y d/dx
), also
x d/dy - y d/dx
+
x
2
+
y
2
=
x
2
+ y
2
+ x d/dy - y d/dx
; therefore the symbols
x d/dy - y d/dx
and
x
2
+
y
2
combine according to the laws of ordinary algebraical symbols, and consequently partial differential equations, which can be put in a form involving these functions exclusively, an be solved like algebraical equations. We shall give some instances of this.
Composition of Solutions of Linear Partial Differential Equations in Two Independent Variables