COR. The equation to an ellipse is r = a.

the focus being the pole, and being measured from that vertex which is the further from the focus ;

.. r = a. √√1 − e2. {1 + 2b. cos 0 + 2b2. cos 20 + ...},

128. To expand 1 (1 e cos 0) in a series of the cosines of 0 and its multiples.

129. To expand (a2-2ab cos + b2)" in a series of the

cosines of 0 and its multiples.

Of the two quantities a and b, let a be the greater.

Now (a2 – 2ab . cos 0 + b2)" = a2n (

b
2. cos
a

b2n

+

a2

Performing the multiplication, we have for the first term,

In a manner similar to that employed in Art. 128, we may expand 1 (a2 - 2 ab cos 0 + b2)".

130. The approximations of Art. 116, 117, to the values of the sine and the cosine of a very small angle, may often be applied to determine the magnitude of Astronomical Corrections, when they are very small quantities.

Ex. 1. If Sin (w - y) = sin w. cos u, where cos u, where y is very small, required an approximate value of y.

Now neglecting the second and all the succeeding terms of the expansion, as being small when compared with 1, we have for a first approximation, (representing by y, this first approximation to the real value of y),