This equation, in the five particular cases just enumerated, 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 Let us now, instead of taking equal angles round the first focus, take a series of eccentric anomalies, in arithmetical progression and let the resulting values of (1) be, as before 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, I when n = (2m + 1). —; 3dly, n = (2m + 1) . (2m+1).; 4thly, n = (6m +1). 377; and 5thly, n = (6m + 2). 77; the fol The Equation {5} may be treated in the the values of „, being ~; ~ + 2; ..... ~ same manner, + 2 ("—1), those of { n π 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 If then, n be even, this is equal to unity, as it evidently ought, 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) 1. If n be even, cos. no cos. n (+4) and, since 1 - e {}, this becomes R......Ra" = I n 1-2λ". cos. nq+λ2” Let = 0, or, let the extremity of one of the R lie in the I n {7,6}. Again, let one only of the R be perpendicular to the axis, and R......Ra”, Here n is of course odd. {7,7}. Next, let one of the R be inclined at an angle, to the axis. 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 I n Lastly, let n be of the form 6m + 2, and 0 = {7,9}· then D & |