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

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*p?y sin 0," sin 0, ... sin 0, 1-pP....(40). But this equation may be satisfied by a function of the form

$ = P8,0,0,... Onni P being a fanction of p only, and afterwards generally 8, a function of O, only. In fact, if we substitute this value of $ in (40), and then divide the result by 0, it is clear that it will be satisfied by the system

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where k,,,,, Mg, dog, &c. are constant quantities.

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

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ht + (− 1) sin Of 20.-*+ (sinone - And) ®ær. 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 n, is of the form

Ors= (sin 0-)'p= (1 -- Malip; where p is a rational and entire function of u = cos fer, and i a whole number. By substituting this value in the general type and making

darv=-i (i+r – 2).................... (43) we readily obtain

0=(1 – u)
To satisfy this equation, let us assume

p= 4,40*** Then by substituting in the above and equating separately the coefficients of the various powers of Mis we have in the first place from the highest

dos =-ele +r – 1)....................(44), and afterwards generally

e-i-2t.e-i-2t - 1

21+ 2 x 2c + p - 2t-3-4 But the equation (43) may evidently be made to coincide with (44), by writing play for , and 27+1) for e, since then both will be comprised in

dest= 'm {) +r – 2}..................(45). Hence we readily get for the general solotion of the system (41),

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p olos le 2w x 21 (6) + 2W +8 -2 3w7

2 x 22%) + 4wton - 30
20.2w – 2 x 2160) +26+8–2.22%) + 2W +8-4
2.4 x 21

plano - &c.}

15 +40 + + 3.2i + 4w +n - 5 where w represents any whole positive number.

Having thus determined all the factors of $, 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'.

llence, 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,

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and P, represents what P becomes when p is changed into

Substituting this value of H, in the equation (25), No. 10, there results since ao = a + h*

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17 K being an arbitrary constant quantity.

Thus the complete value of V for the particular case considered in the present number is V = PO,0,... ,. KP, | -_handh

........... (47), P: (a's + haji

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

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where P', ', ,', &c. are the values which the functions P,

,, ,, &c. take when we change the unaccented variables fun, ... f, into the corresponding accented ones &', 5, ... E, and

n-8+1.n-8+3 ...... n-8 +20 - 1 I n + 2i + 20 -1.n + 27 +20 + 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 is).

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

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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

H = k'a'A“. aSaal-roda (a* – a'), ; provided we make the variable go 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

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