338 
MR. LUBBOCK’S RESEARCHES 
d an sin A / d R 
d t /x 
sin A /d 7?\ m i a~n \/ 1 — e- f r] cos A/ . r cos A — r/ cos A/ ") 
1 — e* ' d * ' p- 1 r /* {r 2 — 2 r r ; ' cos (A — A ; ') -f r y 2 }^ j 
/ 
and since ra d / = — d y, ■ 
d e 
r (cos A + c)/ 
m t ar V I — e~ 
ll o 
i* V l - <?* ' 
vdA / 
P- 
e d tar 
r sin A 
(iE\ 
, m, a r */ 1 — e 
do ~ 
1 
i-H 
> 
\d A/ 
"T 
P' 
d e — 
d rs 
r 4 
d vs C 
1 _ f r y ' sin A,' r sin A — r/ sin A,' | 
1 r / 3 {r 2 — 2 r r/ cos (A — A/) + r/ 2 }^ J 
, 2 ) 
+ 1 
1 . d^ 
\ Smv dl 
9 n 9 
d a = - — d R 
and since r 2 d X = a (1 — e 2 ) d t = a 1 — e 2 r d v 
d < , c v' 1 — e 2 r cos A / d 
do T“ £ Vd s) ~ 0 
d y . a v' 1 — e 2 r sin A /d 
dT + ^ l d77 = °' 
The last six equations serve to determine the perturbations of a comet. 
Let (A e) n be the variation of any element e during the variation A v of v 
at any given epoch n, neglecting the square and higher powers of A v, 
If the values of (A e) be calculated for the epochs 0, 1, 2 , . . m corre- 
sponding to the values v, v + i A v, u + 2 i A v, & c., differing from each 
other by i A v, then the whole variation (S e) of v corresponding to the varia- 
tion i A v of v 
= i {(A c) 0 + (A e) l + (A 
+ '4-’{( A (*«).} - {(**«). - (A««),} + &c. 
= i{d 
-:) + G-:) . . . . 
L vu 
V/ Q \U U/ J 
. i — 1 
f /d e\ /d <?\ ") 
+ 2 
( ( d o) m ~ (d^) 0 / 
A v - 
(*-!)(»'+ 1) f A (de\ 
12 i \ ^ \d vjm 
