of Comeiary Orbits. . . ,-..!.' 187 



; dV_J±_ ' "' - r 



tie ~2qV' 



dV = fije—1) 



dq 2q'V[ ' 



dO_ y^|tW 



de ~~ 211 sin (9 cos 0(2? + R^i/ 

 ■ ' ' ^ - (l+g)y-Rsin g 



<fy ~~R8in0cos0(2gH-Re-^l)" 



To simplify the last two expressions, substitute for R sin 2 # its 



value 



2q + lie-l 

 and we have 



d0 = Q 2 (q-R) 



de ~ K sin (9 cos 6(2q + Re- if 



d0_ q( e +1)(q + 'Ri^l) 

 dq~~n sin 6 cos 0(2? + Re^l) 2 ' 



whence, by substitution and reduction, 



V sin cos -r- • -, -, r-)Ae Aq=-— - — ==-. Ae & Q . 



\de dq dq de) l 2R(2g + Re-l) % 



Putting <? = ! + € and leaving out the factor — , this becomes 



Ae.&q. • . . . . (5) 



2q + U.€ 



The integral of this with respect to eis 



~{2q + K.e+(R-2q)\og(2q + KG)}Aq, . (6) 

 and this, taken between the limits and e, is 



l{ll.e+(R-2 ? )log(l+|)}Ay . . (7) 



= l[e+ log (i+-~)j a q ver y nearl y> 



= -log(l+— e ) Ay nearly; .... (8 



.*.. the number of comets which have perihelion distances 



02 



