188 Mi*. A. S. Davis on the Probable Character 



lying between q and q + Aq and excentricities lying between 1 



and 1 + € is proportional to log ( 1 -f- ~ JAg. 



This expression has been obtained on the assumption that at 

 the distance R there are as many comets with one velocity as 

 with another. Taking the average velocity of comets at a dis- 

 tance R to be 1*6, this assumption may be taken as roughly true 

 for all velocities between 1*6 and 0, and for velocities not very 

 much greater than 1*6. We will assume that it is true for all 

 values between 3 and 0, and that there are no comets with velo- 

 cities greater than 3. The fact that there are many comets which 

 have their perihelion positions nearly in the direction of the sun's 

 motion, shows that there are comets whose independent velocity 

 in space before their attraction to the sun is greater than the ve- 

 locity of the sun, or greater than 1*6. Hence there are comets 

 whose relative velocities at a great distance with respect to the 

 sun is greater than double the velocity of the sun, or greater 

 than 3-2. 



From the formula (3), the value of e for a velocity 3 at the 



9 



distance R is, when R is very large, -q. With our present units 



9 



of time and space jjl =4<ir*, and therefore e= -r—^q ='23 x q ; 



and the whole number of comets whose perihelion distances lie 



between q and q + Aq is proportional to log (l + *23x o")A^ 



The proportion of the whole number of comets to the number of 

 comets with excentricities less than e is 



lo g (l + -23xJ):lo g (l + ^> 



The greater R is assumed to be, the nearer is this ratio to unity, 

 and consequently the smaller is the average excentricity of all 

 the comets having perihelion distances between q and q-\-Aq. 



Hence the average excentricity of cometary orbits with any 

 given perihelion distance q depends upon the distance at which 

 we assume that the directions and velocities of comets are alto- 

 gether independent of the direction in which the sun lies with 

 respect to them. The reason for this may be explained thus. 

 Suppose at a distance R the motions of comets are independent 

 of the position of the sun. Describe about the sun a sphere 



with radius R and another with radius Ynnf\- ^ tne come ^ s 



which enter the larger sphere, those which have a very small ve- 

 locity will have their directions most deflected towards the sun. 



