184 Mr. A. S. Davis on the Probable Character 



This follows also from another consideration. The average 

 velocity of comets when at a great distance from any stars, with 

 respect to any one star, must be as great as their average velocity 

 with respect to any other star. As the average velocity of the 

 stars is probably greater than, or at least as great as, the sun's 

 velocity in space, the average velocity of comets at a great dis- 

 tance with respect to the sun must be at least as great as the 

 velocity of the sun in space. This velocity is, according to 

 Struve, 1*6 radius of the earth's orbit per annum; and this velo- 

 city at a very great distance gives for a comet whose perihelion 

 distance is equal to the earth's distance from the sun an orbit 

 whose excentricity is 1*06, an excentricity much greater than any 

 which has yet been calculated. But the average velocity at a 

 great distance of those comets which come near enough to the 

 sun to be observed from the earth will not be the same as the 

 average velocity of all comets at a great distance. This is easily 

 seen from the consideration that, of two comets which have ex- 

 actly the same direction at a great distance, that which has the 

 smaller velocity will also have the smaller perihelion distance. 

 On this account the average velocity at a great distance of those 

 comets which come within a sufficiently small distance of the 

 sun to be observed from the earth will be less than the average 

 velocity of all comets at a great distance. On the other hand, 

 supposing that there are as many comets in space moving with 

 one velocity relatively to the sun as with another (which suppo- 

 sition will within certain limits be approximately true), the 

 number of comets whose velocity at a great distance is V which 

 within a given time come into the sun's sphere of attraction will 

 be proportional to V, and thus the average velocity at a great 

 distance of those comets which within a given time describe an 

 orbit about the sun will on this account be greater than the ave- 

 rage velocity of all comets. For these reasons the average excen- 

 tricity will differ from 1*06. 



It is my object in the present paper to calculate what is the 

 probability that a comet which approaches near enough to the 

 sun to be observed from the earth will have an excentricity differ- 

 ing by any given amount from unity, and, by comparing this 

 theoretical probability with the facts derived from observation, to 

 draw conclusions as to the probable character of cometary orbits. 



Let us suppose a sphere described about the sun with radius R of 

 such a magnitude that the sun's attraction will not sensibly influ- 

 ence the directions and velocities of those comets which are near its 

 surface. Leaving out of consideration for the present the existence 

 of the fixed stars, this will be the case only when R is taken so 

 large that the velocity which a comet has acquired from the sun's 

 attraction in coming to the distance R is so small that it may be 



