274 CELESTIAL DYNAMICS. 



It follows from this formula that the smaller 2 a (or th 

 major axis of the orbit of a planetary body) becomes, the 

 less will be its velocity when it reaches the sun. This velo 

 city, like the major axis, has a minimum ; for so long as the 

 planet moves outside the sun, its major axis cannot be shorter 

 than the diameter of the sun, or, taking the solar radius as a 

 unit, the quantity 2a can never be less than 2. The smallest 

 velocity with which we can imagine a cosmical body to arrive 

 on the surface of the sun is consequently 



or a velocity of 60 geographical miles in one second. 



For this smallest value the orbit of the asteroid is circu 

 lar ; for a larger value it becomes elliptical, until finally, with 

 increasing excentricity, when the value of 2a approaches in 

 finity, the orbit becomes a parabola. In the last case the 

 velocity is 



or, 85 geographical miles in one second. 



If the value of the major axis become negative, or the 

 orbit assume the form of a hyperbola, the velocity may in 

 crease without end. But this could only happen when cosmi 

 cal masses enter the space of the solar system with a pro 

 jected velocity, or when masses, having missed the sun ,3 sur 

 face, move into the universe and never return ; hence a ve 

 locity greater than -G- can only be regarded as a rare excep 

 tion, and we shall therefore only consider velocities comprised 

 within the limits of 60 and 80 miles.* 



The final velocity with which a weight moving in a 



* The relative velocity also with which an asteroid reaches the solar 

 surface depends in some degree on the velocity of the sun s rotation. This, 

 however, as well as the rotatory effect of the asteroid, is without moment, 

 and may be neglected. 



