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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\


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.



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

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