^4^ Mr, Ivory on a new Method of deducing 
7. If we put 
J = — {sin. 
rt) t ' ^ tan.; 3 
then the equation (9) will become by substitution 
<r" COS. x° . ^ . |jL _ . R° cos. (e°— n ) : 
and this value being substituted in the first of the equations 
(8), we shall get 
<rp cos. h -j- <rp' cos. h* = cos. X° . {cos. 
But, by putting cos. tan. x° for sin. x°, and then dividing by 
sin. i, the third of the same set of equations will become 
(t'p sin. h 4 - (Tp' sin. h' =: • cos. x® . 
' • ” ' sm. 1 
Now if we make 
: tan. 
tan. u = 
sin. i cos. (c® — n) — 
then, by subtracting the two equations just found, after having 
multiplied them by sin. u and cos. w respectively; we shall get 
o-'p sin. (ro — h) — o-p' sin. (/i'— w) = 0 
whence 
/ sin. {ci—b) (/ 
P P ^ sin {b — w) * T* 
/ 
The value of ~, obtained by substituting the series that a-* 
and (T stand for, will be as follows, viz. 
cr' r y T* — t'* \ 
T — T * 1 ^ 6r^ i • 
In this formula 
T — 'T 
6r“* 
is evanescent when r == r'; and in all 
cases it is a very small quantity; because the intervals of time 
between the observations made use of for finding a comet's 
orbit, should in no instance be extremely unequal. If then 
t / 
we suppose ~ we shall get these two formulas, viz. 
