134 
Mr, Ivory on a new Method of deducing 
To determine the constant quantity in this equation, it is to 
be observed that the first term is the square of the velocity in 
. the orbit : if then we suppose the orbit to be an ellipse of 
which/ is the mean distance ; /. t the eccentricity ; and v and u' 
the velocities at the perihelion and aphelion distances; the 
foregoing equation will become at those two points of the 
orbit, 
V — 
= const. 
■■■■ , = const. 
and if we multiply these by (i — and (i + e)* respectively, 
and then subtract them, we shall get 
( 1 — e — u'" . ( 1 + 6 )^ + / “ ”■ 4^ ^ const. 
but U./(l— £) _u'./(l4- I ) = 0 ; for these quantities are 
respectively the doubles of the sectors described at the peri- 
helion and aphelion in the time denoted by unit : hence - 
= const. : the foregoing equation will therefore become, 
2,1 
, ^ - — - 4 - "7 — 0 . 
I 
7 
Again let the equations (i), multiplied by x,y, % respectively, 
be added to the last equation ; then observing that xddx -j- 
yddy + zddz -j- dx^ + + d%" = 3 we shall get 
X 
2 
dd 
dr 
. r* 1,1 
r f 
0 . 
o d 
Let denote the value of r, and 9 .p that of at the epoch 
from which the time is reckoned ; and further let the value 
of be assumed in a series with indeterminate coefficients, as 
follows, viz. 
^ -f- SLpr -j- At* -|“ &c. : 
