a first Approximation to the Orbit of a Comet, 135 
then — = ~ &c. ; and if these values be substituted in 
f fO fOi ' 
the Foregoing equation the coefficients A, B, &c. will be readily 
determined ; which being done, we shall get 
It is next necessary to integrate the equations (1). I begin 
with seeking an incomplete integral of this form, viz. 
5 == 1 + Ct“ + Dr^ &c. : 
and, having substituted in the first of the equations (1), ^ for x; 
and, for its value taken from the preceding series for r ; 
I get, 
by means of which equation, the coefficients in the series for 
s, are easily found : then 
^ = ( 2 )- 
I now seek another incomplete integral of this form, viz. 
(T^rr-j-E.T^-^F.T”* &c. : 
and, having substituted a- for x, I get, 
whence the coefficients of the series for <r will be readily 
found ; then 
«• = •>■ — + (-3). 
The complete integrals of the equations ( 1 ) can now be 
obtained, and they are 
x = sx + <r 
y = sy° + 0-% 
z^sx” + a- . 
