274 CELESTIAL DYNAMICS. 



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

 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 2 a 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 



^=445,750, 



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's 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. 



