IN PHYSICAL ASTRONOMY. 
329 
= — 
r/ Vl 4 7 ^ ^ 1 + 3 cos ( 2 x> “ 2 V) + 1 2 5 5y COS (X' — A/) — 2 5 s } 
r '3 
+ {3 (1 - 4 s 2 ) cos (X' - X/) + 5 cos (3 X' - 3 X/)} 
r' 4 
+ g^s {9 + 20 cos (2 X' - 2 X/) + 35 cos (4 X' — 4 X/)} 
/d R\_ m $ cos(x v — X/) + ss, r (1 + s 2 ) — r] {cos (X v — X,') + ss,} -» 
\dr/ ' f r/ 2 (1 + sfY {r' 2 (l + s 2 ) — 2r'r/ {cos(x' — X,') + 55 ,} + r,' 2 (l + s/)}^ J 
/d i? \ f r sin X' — X,' 
idlj-- w 4 r; 2 (i+^~{ 
r' r] sin (x' — X ; ') 
f r ' s i 
m. < — — 
l ?’,' 2 (1 + s, 2 
+ 
r' 2 (l + s 2 )— 2 r'r,' {cos (X v — X,') + ss,} + r,' 2 (1 + s i 
r* s — r r, s i 
/)}*} 
r i 
^p} 

\ d s / 1 \ r 'z ^ i _j_ 5 2j4 ~ r | r ^ (i + s 2 )— 2 r r,' {cos (x' — X,') + ss,} + r,' 2 (1 + s/ 
Let P be the place of the planet m, S the place of the sun, S N the intersec- 
tion of the orbits of m and m p S L the line from which longitudes are reckoned, 
P,\ the projection of P, upon the plane of the orbit of P ; then if the plane x y 
coincide with the orbit of m, S P = r, P S L = A, N S L = v, P, S N + N S L 
= a„ p;sl = x;, s p; = r ,\ s p , = r, = *•; (i + ^ 
f SP x SP, cos PSP, 1 
=. m A 
R 
SP, 3 
+ 
{S P 2 - 2 S P x S P, cos P S P, + S P 
s p - s p; cos p s p; 
■/>*} 
{s p 2 - 2 s p x s p; cos p s p; + s p, 
SP 2 - SP x S P, cos PSP, 
/}*} 
Y d ^ f sp;cospsp; 
Vdr/— m ‘\ SP, 3 
/d _ f SP x S P, cos PSP, 
\dr)— m ‘\ SP, 3 + {SP 2 -2SP x SP,'cosPSP; + SP, 
fdR\_ f SPx SP,'sinPSP,' SP x SP/sinPSP,' 
I i.l — — 771 . n ? — 
rA 
\dx/ 
V 
SP, 3 
\ d s / 
1 
{s p 2 -2 s p x s p; cos p s p; + s p, 
s p x p, p; 
Z 3 }"} 
SP, 3 
VI 4- } 
{s p 2 - 2 s p x s p/cos p s p; + s p; 
cos (A — A,) = cos (A — v) cos (A, — v) + cos i sin (A — v) sin (A, — v) 
= cos 2 ~ cos (A — A,) + sin 2 ~ cos (A + A, — 2 v) 
tan (A, N — v) = cos t tan (A, — v) 
. . , x cos t tan (X, — v) 
Sin (A. — v) = t ^ — ri 
v 1 ' (l + cos 2 » tan 2 (x, — v))% 
2 u 2 
