138 Mr. Ivory on a new Method of deducing 
(t'p cos. a cos. C + a-p' cos. a' cos. c' — C')S. A° cos. c" = 
— (t'R cos. e — ctR' cos. e' + o-"R° cos. e° 
c"p cos. A si'i. c -f- crp' cos. >! sin. e' -j- cr''p° cos. A^sin. = (7) 
— a'R sin. e — o-R' sin. e' + o"^'R° sin. 
(T*p sin. A + (Tp sin. a' — <r"p® sin. A® = 0. 
These equations are universally true of all orbits; and, in 
order to solve the general problem of finding a planet's orbit, 
they require only to be properly discussed with the view of 
obtaining the values of p, p", p' by means of formulas sufficiently 
exact and commodious in practice : but, at present, I confine 
my attention to the case of the comets moving in parabolic 
orbits. 
4. We may apply the equations (5), which are general for 
all orbits, to the orbit of the earth : let pc, pc', pc" represent 
what (T, (t'. O'" become in the case of that orbit ; that is ( neg- 
lecting quantities of the fourth and higher orders), let (Equat. 
2, 3> 
/ 
T 
6R0J 
|x"= (t + t') — 
(T-fTpS ^ 
6Ro* 
then, Equat. (5), 
pc'R cos. e -j- pcR' cos. e' — [x'^R® cos. = 0 
pc'R sin. ^ [J.R sin. e' — pc"R® sin. e° = 0. 
On account of these formulas, we have 
«-'R cos. e -j- (tR' cos. e' — o'"R° cos e"^ = 
(cr' — (x') R cos. ^ + (<r — pc) R' cos. e' — (a-" — pc") R® cos. e®: 
and^ by substituting the series that o-, o-', o-"; pc pc , pc"; stand 
for, in the right hand side of this equation, and neglecting 
quantities of the fourth and higher orders, we siiail get 
