a first Approximation to the Orbit of a Comet, 333,' 
through which the earth descends towards the sun in a second 
of time; and is the velocity produced by the solar force ; 
therefore the direct velocity towards the sun produced by the 
same force acting on the comet at the distance r, is ^ ; and, 
this being decomposed into the equivalent velocities parallel 
to the coordinates, these velocities will be — , ^ re- 
spectively, which are the actual diminutions of the velocities 
v/ith which the coordinates increase ; but if dr represent the 
fluxion of the time, the diminutions of the velocities with which 
the coordinates increase, will be represented by — 
— therefore we shall have these equations, viz. 
m*x ddx ^ ddy ^ ddx. 
Now if we put T to denote the mean motion of the earth, 
proportional to the time, and estimated by the arc of the circle 
whose radius is the mean distance, or unit ; then will de- 
note the same thing that is signified by in^dT^ in the foregoing 
equations, which will thus become. 
ddx . 
ddy 
d7 
4-^ = 
• r* 
ddx . z 
dr^ r* 
0 
0 
0 , 
( 0 . 
Multiply these equations by 2dx, ^idy, <id%, respectively; then, 
because ^xdx + *iydy + ^%d% = ^rdi\ we shall get by adding 
them 
2dxddx-\-zdyddy-\-zdzddz , zdr 
and, by integrating, 
dx* -f dy'^ -f dz^ 
77 
~ = const. 
r 
