of Comet ary Orbits, 185 



neglected in comparison with the whole velocity of the comet. 

 We will take R so large that this may be the case for all comets, 

 except those whose velocities at R are very small indeed ; and we 

 will for the present consider that even these comets with very 

 small velocities are as likely to have one direction as another. 



Let us fix our attention upon those comets which enter the 

 sphere through any small area. About the centre of this small 

 area describe a sphere, and let r, 6, <j> be the polar coordinates of 

 any point on this sphere, being the angle which the radius 

 through the point makes with the radius through the point op- 

 posite the sun. The probability that a comet will have a direction 

 parallel to any of the radii which pass through the small element 

 of the sphere r 2 . sin 6 . Ad . A<£ is proportional to this area. 

 Hence the probable number of comets having directions parallel 

 to radii through r 2 sin 6 . A0 . A<j> which pass within a given time 

 through an area perpendicular to their direction is also propor- 

 tional to r 2 sin . A0 . A<£ ; and therefore the probable number 

 which will pass in a given time through the small area of the 

 sphere about the sun, which area is inclined to their direction at 



an angle - — 0, is proportional to r 2 sin cos A0 . A$. 



Hence the probable number of comets which will pass through 

 the area in a given time and are inclined to the sun's direction 

 at angles lying between 6 and# + A0 is proportional to 



r 2 . sin 6 . cos 6 . A0. 



We have not yet considered the velocity of the comets which 

 enter. Supposing that there are as many comets with one velocity 

 as another, the probable number of comets which in any time 

 enter any area and have velocities lying between V and V + AV 

 will be proportional to V. AV. Hence the probable number of 

 comets which have velocities betweenVand V + AV and directions 

 inclined to the sun's direction at angles between and + A0 

 which pass in any time into the sphere is proportional to 



Vr 2 sin0cos0A<9AV (1) 



We proceed now to find the probable number of comets having 

 perihelion distances lying between q and q-^-Aq, and excentrici- 

 ties lying between e and e + Ae. 



We have the following equations between V, 6, e, and q : — 



l_ e ___(___) .... (2) 



(see Tait and Steele's 'Dynamics/ p. 91), and 



vHh~) (*> 



Phil. Mag. S. 4. Vol. 40. No. 266. Sept. 1870. O 



