38 G. Hinrichs on the Density, Rotation, and Age of the Planets. 
resolving the forces into a radial and transversal component wé 
obtain the general equations’ of motion: ‘a? 
d%r d0\2 B " 
7a—"( a) =— atR sin, ] 
dir? a (1) 
, a =—R cos 
wedi cua cs 
We might integrate (1) by the methods of approximative 
integration, as Laplace has done in a similar case; or we migh 
integrate for R=0 and use the variation of parameters. This 
latter method would be much more appropriate than the first; 
yet we think that a simple successive approximation is fully 
exact enough and at the same time so much more elegant and 
easy that we will prefer it, considering uniformity to be a true 
element of any investigation. We therefore directly aim at just 
that degree of approximation which observation enables us to 
test—and also thereby keep this paper within the range of almost 
every student of the calculus. 
rst approximation gives Kepler’s laws as the integral of 
(1) re R=0, representing the motions in vacuo:* i. e. respect 
ively : 
~ 1+-ecos 0’ ae 
SDS SS oi @) 
~~ T2~a(1 =e?) 
I. The orbit is an ellipse ; 
II. The radius vector of any planet describes equal areas in 
equal times ; 
ITf. The mean radii vectores (mean distances) of the several 
planets describe in their mean motion equal solids in equal times. 
* These formule are easily obtained as stated; compare Price, Infinit. Calculus, 
vol. iii, Art. 297, formula (174), remembering that P = gravitation =— =. Q=0 
“SSE g d 
sin 7= Ze? 87= en and as a simple identity 
do 
dr do, d90__ 1 /2rdr do d29 a(r a) a 
2—-— a 8 baie hy | 
Wie tea a ti 08 +e. 
* The first and second are of familiar form; the third i. hae 
Caleulus, vol. iii, Art. 331, where A is near his = may be found in Price’ 
: be analogous to the 
mean motion of the individuals I found this form whilst searching for the harmony 
between these two laws, and used it as early as 1857 in publie lectures on the pria- 
