16 



MEMOIRS OF THE NATIONAL ACADEMY OF SCIENCES. 



25. The dotted curve in the several figures represents those values of d and /( for which the 

 total change of direction in the relative orbit is 10°; that is, a=85°. It is thq,t curve whose equa- 

 tion is A tan 85o=B, or d^+¥ sin^ ^^A^ tan^ 8oO. It is therefore an ellipse whose center is the 

 origin of coordinates, and it is similar in each figure to the isergonal ellipses. 



26. Eypotheses abovt the parahoUc conietary orbits. — It will be convenient to make two assump- 

 tions about the distribution of the parabolic comets, and the distribution of the goals of their 

 motions. There seems to be no very well marked relation between the ecliptic, or to speak more 

 strictly, the invariable plane of the solar system, and the known parabolic cometary orbits. The 

 following two assumptions do not seem likely, therefore, to introduce any very serious error into 

 our reasonings. 



If about the sun as a center a sphere ^ be described with an arbitrary radius r, it will be 

 rt.s'SM}»ef7 that near the surface of ^, space is filled equably with comets. We may express this 

 by supposing that in each cubic unit of space near f , there are at each and every instant n comets. 

 As the orbits are all assumed to be parabolic, the n comets have a common velocity v. 



Fig. 12, 



It will be furthermore assumed that the directions of the comets in each cubic unit of space 

 near ^ are at random, that is, that the quits and goals of the comet's motions relative to the sun 

 are distributed equably over the surface of the celestial sphere. 



27. Number of comets entering ^. — If about a normal to ^ as an axis there be described two 

 cones cutting the celestial sphere in two small circles distant from the point where the normal 

 meets the celestial sphere ip and ip + dtp^ then of the n comets there will be ^nsinipdip comets 

 whose quits are between the two circles. Each of these comets will move perpendicularly to the 

 spherical surface ^ with the velocity v cos ip. Hence in a unit of times i«?icos ipsimpdip comets 

 will cross a unit of the surface % going toward the sun. The total entering the sphere in the unit 

 of time will be this number multiplied by the number of units in the surface of ^, or 



2;r»-^ 



if- 



cos 'psu\ipd>/-z=TTnvr^. 



