IN PHYSICAL ASTRONOMY. 
39 
d , + { ( , + ,> } (<f) -,>.(£)- (31) (&) } a V = » 
(see Phil. Trans. 1830, p. 334), serve to verify some of the theorems of Newton 
in the third volume of the Principia. 
In fact 
R= -ElL: j 1 +3 cos (2A-2A,) -2s°-j 
(I+5 ' !) (l7)="'' s(1 + s ’ ) ^ 
' S = +3“S(2A-2X,)-2 S *} S 
( d R\ 3m.r 2 . /0 0 , N ds 
r? j = ^ ;r s,„(2x-2A, ) 
(*' 
-v) 
neglecting s 3 , 
, . m, r* sin (A — v) J . N . sin (A — y) f ' , „ . n , . 1 
dv + ~ — Wr» -j sin ( x_v )+ — ^2 l + 3cos ( 2X ~ 2x /) ) 
— ~ cos (X — y) sin (2 A — 2 A y ) j d A = 0 
d v + m ‘ r S ^ X ~ ^ | sin (X — v) + 3 cos (X — A,) | cos (X — X ; ) sin (X — y) 
— sin (X — X,) cos (A — y) j — sin (X — y) | d X = 0 
d y = 3 ' m ‘ r sin (X — y) cos (X — A,) sin (X ; — y) d X' 
h~ r 3 
_ 3 ?», a 1 gin ^ ^ cos ^ sin ^ _ x ) d x nearly 
= — - [ _ . sin (X — y) cos (X — X ( ) sin (A, — y) dX 
o9*5 / 5 
Which agrees with the result of Newton, Prop. Lib. 3. “Est igitur velocitas 
nodorum ut IT x PH X AZ, sive ut contentum sub sinubus trium angulorum 
TP I, PTN et STN Sunto enim PK, PH et AZ prsedicti tres sinus. 
Nempe PK sinus distantise Lunse a quadratura, PH sinus distantise Lunse a 
nodo et AZ sinus distantise nodi a Sole, et erit velocitas nodi ut contentum 
PK X PPI X AZ.” 
Similarly 
d ( = 3 m j r t C | ° S - i sin i cos (X — y) cos (X — A,) sin (A, — y) d X 
