λιπ

λεπ

λεπ

GG-1+HG-1+KG"-1
GG-G-2+HG-G-2+KG"-1G-2

+ GmHm-1G'm-2+ HmH'm-1G'm-2+K„H"m-2G'm-2

+GmK-1G"-2+HK'-1G"-2+KK"m-1G" -2
GG-1G-2G-3+HG-1G-G-3

+ KG"-1Gm-2Gm-3+GmHm-1G'm-2Gm-3
+ H„H'‚-1G'm-2Gm-3+K„H”m-2G′mGm-3
+ G„K„-1G”„-¿G„-3+H„K'„-1G”„–¿G„–3

m-2

+GH-H'-G'm-3+HH'H'G'm-3
H„H',

+KH"-H'-G'-3+K,H"m-H'm-2G'm-3

+GK-H"-2G'm-3+ HK'm-1H"m-G'm-3

+ GG-1K-2G"m-3+HG'-1K-G"-3
+KG"-1K-2G"m-3+GH-K'-G"-3
+HH'm-1K'm-2G"m-3+K„H"m-1K'm-¿G",

m-3

+GmKm-¿K”‚¬¿G"m-3+H„K'm-1K′′m-¿G′′m–3

+KK"-K"-G"m-3

-Write

Hence we obtain the following rule for the determination of X, :down the term GmGm-1Gm-2. . . . . G-r. We may substitute H and K at pleasure for G anywhere except in the last factor, which is always G. Whenever we put H for G, the succeeding letter is to receive a single accent; whenever K for G, the succeeding letter receives a double accent. The aggregate of all the terms thus formed will be λ,, and we may of course obtain similar expressions for Mrs &c.

we shall find the above equations satisfied; and consequently the last investigation gives the law of the formation of the remainders. Each remainder will of course be subject to the three conditions already exhibited.

These results point out the foundations on which symbolical division, as applied to non-linear functions, must rest. We have confined our attention to external division, as more particularly applicable to these functions. When a non-linear equation is proposed for reduction, we must ascertain whether it admits of an external factor by employing the method of division as already explained.

"On the Calculus of Symbols.-Fifth Memoir. With Application to Linear Partial Differential Equations, and the Calculus of Functions." By W. H. L. RUSSELL, Esq., A.B. Communicated by Professor STOKES, Sec. R.S. Received April 7, 1864*. 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 func tional 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.

laws of ordinary algebraical symbols, and consequently partial differential

*Read April 28, 1864. See Abstract, vol. xiii. p. 227.

and 2+y combine according to the

equations, which can be put in a form involving these functions exclusively, can be solved like algebraical equations. We shall give some instances of this.

where

dx

dw

y f(x, y)+(x2+y2)w'

c2-x2)

* (daf (x, No c2 = x2)

'dxf(x, √ c2 —x2) =F(x, c) ;

2

+ βε

-c2 sin-12

sin-12

√x2+y® F(x, √x2+y3)

is an arbitrary function. We shall denote this expression by x(x, y), whence we have for the determination of (u),

which may be treated as before.

{

In order to find the most general form of equation to which the symbols

-y dx

and x2+y2 give rise, we must determine the expansion of

to avoid the

will consist of all the terms of the

d a