de, d,... d=p1 sin 0, sin 0,**... sin o, do do, do,... de...»›

Proceeding now in the manner before explained (No. 8), we obtain for the equivalent of (39) by reduction

But this equation may be satisfied by a function of the form

P being a function of p only, and afterwards generally ✪, a function of 8, only. In fact, if we substitute this value of & in (40), and then divide the result by 4, it is clear that it will be satisfied by the system

where k, A, A, A,, &c. are constant quantities.

In order to resolve the system (41), let us here consider the general type of the equations therein contained, viz.

Now if we reflect on the nature of the results obtained in a preceding part of this paper, it will not be difficult to see that O is of the form

O,, = (sin 0 ̧‚)* p = (1 -- μ3)3p;

where p is a rational and entire function of μ = cos er, and i a whole number.

By substituting this value in the general type and making

Then by substituting in the above and equating separately the coefficients of the various powers of μ, we have in the first place from the highest

λr = − e (e + r− 1).............................................. ..(44),

But the equation (43) may evidently be made to coincide with (44), by writing ) for i, and +) for e, since then both will be comprised in

(41),

Hence we readily get for the general solution of the system

where μcos e, and i) represents any positive integer whatever, provided is never greater than +)

Though we have thus the solution of every equation in the system (41), yet that of the first may be obtained under a simpler form by writing therein for λ its value - deduced from (45). We shall then immediately perceive that it is satisfied by

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

d'P-1-np dP (i) (+82) k

0=

+

dp2 p (1-p2) dp

+

P

which is satisfied by making k-λ,- (¿'() +2w) (¿()+2w+n−1),

and

+

P = poten

20.2w ·2 × 2i(1) +2w+s · 2.2i+2w+s-4
2.4 × 2i(®) + 4w + n + 3. 2i + 4w+ n − 5

where a represents any whole positive number.

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

since in the present case we have generally a,' a'.

=

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

entering into is likewise constant. Neglecting therefore this factor as superfluous, we get for the particular value of H,

and P. represents what P becomes when p is changed into a

Substituting this value of H, in the equation (25), No. 10, there results since a a2+ h2

=

H = K.P
K.P.

hdh

.(46),

"P" (a'3 + h3)

K being an arbitrary constant quantity.

Thus the complete value of V for the particular case considered in the present number is

and the equation (27), No. 11, will give for the corresponding value of p',

where P, O,,,, &c. are the values which the functions P, ,,,, &c. take when we change the unaccented variables ,,,, ,,... into the corresponding accented ones, §.', ... §.',

n-s+1.n - -s+ 3...... n — 8+20-1
n+2i+2w −1 . n + 2i + 2w + 1 ...... n + 2i + 4w — 3 '

or the value of P when p=1; where as well as in what follows

i is written in the place of "".

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

+ {i (i + s − 2) a” — (i + 2w) (i + 2w+ n − 1) a'} H,

and the particular integral here required is that which vanishes when his 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 k' may be determined by comparing the coefficient of the highest power of a in the expansion of the last formula with the like coefficient in that of the expression (46), and thus we have

k' = Ka'12» (− 1)∞ 12 + 2i+2w −1.n+2i+2w+1... n+2i+4w−3

2.4.6 20

...

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

n+2i+ 2w −1.n+2i+2w+1...n+2i+4w-3

2.4.6...20

In certain cases the value of V just obtained will be found more convenient than the foregoing one (47). Suppose for instance we represent the value of V when h=0, or a=a' by V。. Then we shall hence get

n+2i+2w-1.n+2i+2w+1...n+2i+4w-3 2.4.6...20

× Ka'+(-1)". A*a* f“ daa1** (a2 — a′′3)

....

(8),