, , /2 , e-l\ 



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



Substituting for V 2 in (2) from (3), we have 



2 <?-l\li 2 sin 2 



> 



or 



f/(l-}-t')--2%sm 2 (9--(e-l)R 2 sin 2 ^ = 0. . . (4) 



To a perihelion distance q and an excentricity e correspond a ve- 

 locity V and a direction at 11 given by (3) and (4). To q-\- Aq 



and e correspond V-f- -=— . Aq and 0-\- -r-.Aq. To q and e + Ae 



correspond V + -7- . Ae and + -j-.Ae; and to q + Aq, e-\-Ae 



i tt . dY . , dY A i /1 . ^ a , $! A 

 correspond V + - 1 — . A<7 + -=- . Ae and + -7- . Aq + -j- . Ae. 



If we take V and as the coordinates of a point referred to 

 rectangular coordinates, then AV. A0 represents the area of a 

 small rectangle whose sides are AV, Ad, and the expression (1) 

 may be written 



V sin cos x rectangle dY .d0. 



So if « be any small area about the point whose coordinates are 

 V, 0, the expression V sin cos . a is proportional to the num- 

 ber of comets for which the values of and V are represented by 

 points within the small area a. Now all comets which have pe- 

 rihelion distances between q and q + Aq, whilst their excentrici- 

 ties lie between e and e + Ae, have directions and velocities at 

 a distance R represented by points lying in the parallelogram the 

 coordinates of whose angular points are respectively 



«r /* /\t dY a a dO , A \ / TT dV . . <*0 A \ 



and 



/„ dV A dV A . 40 A <*0 A \ 



The area of this parallelogram being 



'd\ t d6__dV dO^ 

 de dq dq 



the number of comets under consideration is proportional to 



v • n nfdV dO dY d0\. ' 



V sin . cos 6 f -, ; 1 r A? . Ae. 



\ ae d dq de J 



Differentiating (3) and (4) with respect to e and q, we find 



dY d0\ A A 



