﻿48 Mr. A. S. Davis on the Probable 



value of e is given by the equation 



E _ SH? . _ 3g 



q~~ € ' ''• 6 ~" R + ^" 

 Substituting this value in the expression above, we obtain 



— log -p^ x Sg= log — x Bq very nearly. 

 sx-t q Q 



Thus, on the supposition that after comets have arrived at a 

 distance R, from the sun they cease to be attracted by the stars, 

 the probable number of comets having perihelion distances be- 

 tween q and q-\-Sq and excentricities less than unity is propor- 



tional to log — X $q. But the attraction of the stars upon 'a 



comet does not cease when it has arrived at a distance R from 

 the sun. The effect, however, of their attraction will be to di- 

 minish the number of comets with perihelion distances so small 

 that they can be observed from the earth ; for their attraction 

 will generally diminish the force acting upon the comet in the 

 direction of the sun, and will cause the direction of the whole 

 resultant attraction not necessarily to lie in the direction of the 

 sun. Thus the comets will have their directions of motion less 

 deflected towards the sun than they would be if they were not 

 attracted by the stars, and consequently will have larger perihe- 

 lion distances, so that fewer will be observod from the earth. 



We may then take log — x Sq as a superior limit to the number 



of comets with perihelion distances between q and q + &q, and 

 excentricities less than unity. We have, therefore, the ratio of 

 elliptic to hyperbolic orbits less than 



log|x8 ? :log(l+-23x J)xS ?J 



a ratio which for moderate values of q does not differ much from 

 unity. 



Hence out of every hundred comets approaching the solar 

 system for the first time and having perihelion distances equal 

 to unity, about fifty will move in hyperbolic orbits, and of these 

 fifty about ten or eleven at least will have excentricities greater 

 than 1*02. For comets with perihelion distances larger than 

 unity there will be even a larger percentage than ten or eleven 

 having excentricities greater than 1*02. 



We are thus led to the conclusion that a large majority of 

 those comets whose orbits are un distinguishable from parabolas 

 are not visitors to the solar system for the first time, because, if 

 they were, the percentage of comets with excentricities greater 

 than 1 '02 would be very much smaller than by theory it should be. 



