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This equation, in the five particular cases just enumerated,

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p(2) ........... p.(2) = a” . ( 1 +×)”. (E)·(E)· · {2,6}

I

.....

n

Our third equation {3}, gives immediately

r(2)....... 7(n) 1-2e". cos. no+e2n

1

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Let us now, instead of taking equal angles round the first focus, take a series of eccentric anomalies, in arithmetical progression

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and let the resulting values of (1) be, as before

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This gives, in the five cases; 1st, when n is any number, and

w=0, (or one of the "') terminates in the first vertex); 2dly,

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when n = (2m + 1). —; 3dly, n = (2m + 1) . (2m+1).; 4thly,

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n = (6m +1). 377; and 5thly, n = (6m + 2). 77; the fol

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The Equation {5} may be treated in the the values of „, being ~; ~ + 2; ..... ~

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same manner, + 2 ("—1), those of

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{

n

π

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As this case, however, is manifestly similar to that of {4,1},

we shall pursue it no farther.

The 6th of the equations, in page 9, offers, however, some results worthy of consideration. By treating it like the rest, it becomes

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If then, n be even, this is equal to unity, as it evidently ought,

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We come now to our 7th Equation, which will afford us results, more complicated indeed, yet equally interesting. By applying the same method of transformation to it, we shall

2(n-1) π

find, (supposing 4, 4, ..... 4 = 4 + —1), to be written for

1 2

n

I

n

❤, and R, R,..... R to denote the resulting values of R)

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1. If n be even, cos. no cos. n (+4) and, since 1 - e

{}, this becomes

R......Ra"

=

I

n

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1-2λ". cos. nq+λ2”

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Let = 0, or, let the extremity of one of the R lie in the

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I

n

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{7,6}.

Again, let one only of the R be perpendicular to the axis, and

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R......Ra”,

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Here n is of course odd.

{7,7}.

Next, let one of the R be inclined at an angle, to the axis.

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I

n

and it is curious to observe, that this expression is the same

function of a, λ, n, as that of {7,7}.

If n be of the form 2m + 1,

R......Ra
a" (1-x2)

I n

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Lastly, let n be of the form 6m + 2, and 0 =

{7,9}·

then

D &

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