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From a graphical representation of these velocities, the velocities 
of sound at 5, 15, 25 KM. ete. were read, and in order to simplify 
the ecaleulation we assumed that for the less curved part of the 
orbit a sufficient approximation was obtained by considering these 
velocities as constant every time over 10 KM. ; 
Starting from some initial zenith-distance of the ray «, we find 
an from sin a,: sina, = V,: Vi; the horizontal projection of the 
sound rays is then given by 2 10 ty ay. 
For the more curved part of the orbit and especially when it 
becomes horizontal, this method, however, would cause too great 
errors. It appeared, that up to the last 5 KM. height a sufficient 
accuracy was obtained by making the calculation by steps of 1 
instead of 10 KM.; hence reading the velocity at each full KM. and 
using the mean value, which practically means calculating 11 steps, 
the first and last of which only count for one half. 
Also this method would cause errors for the last kilometers as 
the tg. approaches o for « — 90°. 
To get a simple approximation here, we used the simplified 
differential-equation of the orbit for the case where the velocity 
of sound at this height varies in a linear way, hence may be 
represented by v= v,—ch. 
If we call z and / the co-ordinates of a point of the orbit, starting 
from the summit of the orbit as origin, then : 
dh v C 
cotg = — ig ee ml ch 
dx Un Vn 
cos a —= V 2ch—e?h? 
(l1—ch) dh 
de = dh tg a = ——_____ 
V 2c'h—ce'h? 
On further inquiry it appears that in this case the orbit is a 
: et, 
circle with a radius — == —. 
c c 
The approximation obtained in this way happens to be closest in 
the vicinity of the ray which reaches the earth at the minimum 
distance from the source of sound, because this ray is inverted in 
the region where the velocity of sound varies most rapidly, and 
therefore the curve of sound-velocities shows an inflexion. 
This method of calculation was also applied to the results com- 
municated graphically by von pem Borne for the velocity of sound 
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