IN PHYSICAL ASTRONOMY. 
39 
d< + 
r 2 cos Ceos (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 
= ji +3 cos (2 A — 2 A,) -2s 2 j 
K d df ) = ”' s(1 + s,) $ 
(1 + s 
' s (a?) = “’ll? { 1 +3 cm(2X-2»,)-2^}, 
( d R\ 3m, f® . /0 ,, d s ,,, . 
d?) = -2fcr sm(2 ^ 2A ' ) dT^'an'Cos^-’) 
neglecting s 3 . 
, . m, r 4 sin (A— v) J • / , N . sin (A — v) _ . 1 
d v + . h ^ 3 -A |sin (A - v) + -A_- L j 1 + 3 cos (2A — 2 A,) j 
— — cos (A — y) sin (2 A — 2 A,) j. d A = 0 
d v + m ‘ -- {sin (A — v) + 3 cos (A — A,) j cos (A — A ; ) sin (A — y) 
— sin (A — A ( ) cos (A — y) j — sin (A — y) j d A = 0 
d v — sin (A — y) cos (A — A ( ) sin (A, — y) d A' 
h~ r ; 3 
_ 3 m , a s j n ^ ^ cos ^ ^ s j n ^ _ v) d a nearly 
= 59 .^ 5 sin (A — v) cos (A — A,) sin (A, — y) d A 
Which agrees with the result of Newton, Prop. Lib. 3. “Est igitur velocitas 
nodorum ut IT x PIT X AZ, sive ut contentum sub sinubus trium angulorum 
TP I, PTN et STN.Sunto enim PK, PIT 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 PH X AZ.” 
Similarly 
d i = —-— sin t cos (A — y) cos (A — A,) sin (A, — v) d A 
