This known abridged notation is derived from the analogy which exists between integrals and powers. As to the use made of it here, the object is to express series, and to verify them without any development. It is sufficient to differentiate under the signs which the notation employs. For example, from the equation v = et13 p (x), we deduce, by differentiation with respect to t only,

which shews directly that the series satisfies the differential equation (a). Similarly, if we consider the first part of the series (X), writing

v = cos (x√-D) & (t),

we have, differentiating twice with respect to x only,

Hence this value of v satisfies the differential equation (a).

We should find in the same manner that the differential equation

gives as the expression for v in a series developed according to increasing powers of y,

v = cos (yD) $ (x).

d

We must develope with respect to y, and write

instead of

dx

The value sin (yD) ↓ (x) satisfies also the differential equation; hence the general value of v is

v = cos (yD) 4 (x) + W, where W = sin (yD) ↓ (x).

and if we wish to express v in a series arranged according to powers of t, we may denote by Do the function

We must develope the preceding value of v according to powers

instead of D', and then regard i as the order

condition; thus the most general value of v is

and

v = cos (t√-D) $ (x, y) + W ;

W= [dt cos (t √— D) ↓ (x, y) ;

v is a function f(x, y, t) of three variables. If we make t = 0, we

the value of v in a series arranged according to powers of t will

The general value of v, which can contain only two arbitrary functions of x and y, is therefore

v = cos (tD3) & (x, y) + W,

and

W

= √

dt cos (tD3) (x, y).

dv

Denoting v by f(x, y, t), and by f'(x, y, t), we have to

dt

determine the two arbitrary functions,

$ (x, y) = ƒ (x, y, 0), and † (x, y) = ƒ (x, y, 0).

ď ď
dx2 dy"

we may denote by Do the function + so that ᎠᎠᏜ

d2 d2
dy

or D'o can be formed by raising the binomial ( +

to the

second degree, and regarding the exponents as orders of differen

d'v dt

tiation. Equation (e) then becomes +D'v=0; and the value

of v, arranged according to powers of t, is cos (tD) & (x, y); for from this we derive

The most general value of v being able to contain only two arbitrary functions of x and y, which is an evident consequence of the form of the equation, may be expressed thus:

х

v = cos (tD) $ (x, y) + [ dt cos (tD) ↓ (x, y).

and if we wish to express v in a series arranged according to powers of t, we may denote by Do the function.

We must develope the preceding value of v according to powers

of t, write(+ 2)", instead of D', and then regard i as the order

of differentiation.

dy

Sat

The following value dt cos (t√- D) ↓ (x, y) satisfies the same condition; thus the most general value of v is

v is a function f(x, y, t) of three variables. If we make t = 0, we

the value of v in a series arranged according to powers of t will

or Do can be formed by raising the binomial

second degree, and regarding the exponents as orders of differe tiation. Egation then becames+D=0; and the val of v, arranged according to powers of t, is cos (tD) & (n, th

from this we derive

de

da

arbitrary functions of x and y, which is an evident emp
The most general value of being able to contain
the form of the equation, may be expressed thus
1=cos (tD) + (x, y) + at cos (D),