38 PROFESSOR G. H. OARWIN OR THE MECHANICAL CONDITIONS OF 
where C is a constant which it will he unnecessary to determine, and where the limit 
€q will he the subject of future consideration. 
Effecting the integration, we have 
8 = — ~ (u^ — f [r)) — U'a^ sin^ e}% between limits, 
= a; [ - /O’)) - /(^)) - sin^ e,}’ ]. 
It is obvious that, if is greater than ry’(r)/(r~ — a^), the square root involved in 
drjdt does not vanish for any value of e; and, hence, we must simply take eg = 90°. 
If, on the other hand, id is less than this critical value, Cg is that value of e which 
makes di'ldt vanish. 
Thus, our formula divides into three, viz. :— 
f 
1 st. id greater than 
o / o 
«“/7- 
8 = 
c 
2nd. id less than 
// 
I - cd\f 
C 
8 1= - - /O’) )t 
f{r) Y" 
I - cdld) _ 
3rd. id less than f [r) ; 8 — 0. 
The ph\ sical meaning of this division is as follows : If we take a station near the 
surface of the sphere, meteorites shot out at all inclinations, even horizontally, reach 
the height of our station ; and, when they are shot out horizontally, e = 90°. If we 
go, however, to a higher region, there is a certain inclination which just brings the 
meteorites at apocentre, where drjdt = 0, to our height ; but those shot out more 
nearly horizontally fail to reach us. Still higher, not even a meteorite shot up 
vertically can reach us, and the density vanishes. 
These results only correspond to a single velocity u ; but, if he the mean 
square of the velocity, the number of meteorites whose velocities range between u and 
u-i~di{ is proportional to (/m.'* Hence, we have to imdtiply 8 by this expres¬ 
sion, and integrate from ic = oo to u — 0. 
Now, the first term of the first form for 8 is the same as the second form ; and in 
the third form 8 is zero ; hence, this first term when multiplied by the exponential has 
to be integrated from id =: co to f {i'). The second term of the first form of 8 has to 
be multiplied by the exponential, and integrated from id = co to/('^’)/(t — U'/r'). 
Now, for the first term ])ut 
* Oskar Meyer, ‘ Die Kinetiscbe Tbeorie der Gase,’ 1877, pp. 271-2. 
