40 A, Hinrichs on the Density, Rotation, and Age of the Planets 
pendulum the resistance is principally proportional to the velo- 
city itself, for greater velocities, to its square, or more exactly 
9(v)=av--be?, 
: 
w, as the resistance under consideration is excessively small, ; 
Now. 
the form of g will be very nearly 
o(v)=0. (6) 
Introducin ng these laws (6) and (@) into (8) the equations (1’) 
for the secon hail sea becom 
d6\ 2 ub 
eo ai) core 
(1") 
The first of these is satisfied by the api orbit of each single 
lution ; the second, si = gl educes to 
revolu e second, since v cos7=-~ >=, I 
(a ) zai? 
which, by the second @) expressing the conservation of the 
areas, mes the simple st 
Op ve (8) 
giving, ifc=C for t=0, and « the base of the natural logarithms, 
e==Ca 
(9) 
By the third law = Kepler, i. e. the third of (2) or a= con- 
stant times c? we hav 
ahd 
a=Ae_ gts (10) 
if A be the value of a for t=0. Thus the distance from the sun 
does not decrease ment but with a velocity proportional to 
the decreasing value of a 
tO _ovae™ Vt ane Soe, el 
If ee the present age, distance a, and « the 
correspondin: te an aces differing from the former by # pe re 
time, then (10) gives 
; i. grid BO esol ql 0”) 
giving the distanes ak x a function of its present Meso s and the 
interval of time from the present. As the unk 
9 enters into », we may instead of (10,) use the 
log eloga——, (12) 
