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+ 2.4x{2>l+r-3}{2»"l+r-5} M ~ J'

where = cos and i(rl represents any positive integer whatever, provided i[r) is never greater than ilr+1>.

Though we have thus the solution of every equation in the system (41), yet that of the first maybe obtained under a simpler form by writing therein for X^, its value — im* deduced from (-15). We shall then immediately perceive that it is satisfied by

In consequence of the formula (45), the equation (42) becomes

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where a> represents any whole positive number.

Having thus determined all the factors of cf>, it now only remains to deduce the corresponding value of II. But ff0 the particular value satisfying the differential equation in H, will be had from <f> by simply making therein

since in the present case we have generally a/ = d.

Hence, it is clear that the proper values of 0V 6r 6a, &c. to be here employed are all constant, and consequently the factor

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The differential equation which serves to determine .ffwhen we introduce a instead of A as independent variable, may in the present case be written under the form

0 = a* (a' - a") ^ + a' {«a' - (e - 1).«"} g

+ [i(i + 9-2)a"- (» + 2a)(t + 2a» + n -1) a'}H,

and the particular integral here required is that which vanishes when h is infinite. Moreover it is easy to prove, by expanding in series, that this particular integral is

provided we make the variable r to which A" refers vanish after all the operations have been effected.

But the constant U may be determined by comparing the coefficient of the highest power of a in the expansion of the last formula with the hke coefficient in that of the expression (46), and thus we have

k-Xa (-1) 270772^ *

Hence we readily get for the equivalent of (47),

w + 2t + 2ol-l.n + 2t + 2w+l...n+2»+4(»-S

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which in consequence of the well-known formula

/;.-* (^-.r-."* x ^tt^t '-.

by reduction becomes

r n+s-n\ r/«+2i+4nl-l\

2r(a+1)r(^±^)

since in the formula (8), r ought to be made equal to zero at the end of the process.

By conceiving the auxiliary variable u to vanish, it will become clear from what has been advanced in the preceding number, that the values of the function V within circular planes and spheres are only particular cases of the more general one (49), which answer to s = 2 and s = 3 respectively. We have thus by combining the expressions (48) and (49), the means of determining Ve when the density p is given, and vice versa; and the present method of resolving these problems seems more simple if possible than that contained in the articles (4) and (5) of my former paper.

ON THE MOTION OF WAVES

IN A VARIABLE CANAL OF SMALL DEPTH AND WIDTH*.

* From the Transaction! of the Cambridge Philosophical Society, 1838. [Read May 15, 1837.]

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