ON THE CAPTURE OF COMETS BY PLANETS. 527 



negative value for the quantity in square brackets we must integrate 

 between the two values of o> corresponding to the zero value of the 

 bracketed quantity.] We may make S the independent variable by the- 

 equations 



sds =y 2 sin to dm, Vq^/ 2 = sv, and 2s cos 6 = 1— s^. 



These give 



N=i:r«m^.[[4@2 - ( ®-''®'y y. 



33. If now we require the number of comets which in each unit of 

 time shall pass the planet in such way as that they shall have after the 

 pas.sage respectively less than one-half, once, three-halves, and twice, the 

 planet's period of revolution, we may place @ = rT', and make T equal 

 successively to \, 1, f , and 2, and compute in each case the value of N as 

 given in the last article. The results are found to be irnmh'^v multiplied 

 severally by the coefficients 0'139, 0-925, 1-876, and 2-943. 



34. By comparing the results of Arts. 27 and 33, and making the 

 assumptions of Art. 26, we have the proposition, that the number of 

 comets which in a given period of time pass their perihelia nearer to the sun 

 than a given planet is to the number of comets whose periodic times are 

 reduced by the perturbing action of the planet so as to be less severally than 

 one-half once, three-halves, and twice, the periodic time of the planet, as 

 unify is- fo the square of the mass of the planet m^dtiplied severally by 0-139, 

 U-Sr2>), 1-H76, and 2-943. 



35. If Jupiter is the planet, m=y^Vo) ^^^ ^^ ™^y express these 

 ratios as 



1 000 000 000 : 126 : 839 : 1701 : 2670. 



That is, assuming the hypotheses of Art. 26, and regarding the planet as 

 without dimension so as to intercept any comets, if in a given period of 

 time a thousand millioti comets come in parabolic orbits nearer to the sun 

 than Jupiter, 126 of them will have their orbits changed into ellipses with 

 periodic times less than one-half that of J^ipiter ; 839 of them will have their 

 orbits changed into ellipses with periodic times less than than that of 

 Jupiter; 1,701 of them will have their orbits changed into ellipses with 

 periodic times less than once and a half tim.es that of Jupiter ; and 2,670 

 of them will have their orbits changed into ellipses with periodic times less 

 than twice that of Jupiter. 



36. Another and perhaps a more important inquiry is this, what 

 effect have the perturbations of the planet in bringing or not bringing 

 the comets to move in the same direction that the planet is moving after 

 the comets have by perturbation had their periodic times largely reduced ? 

 For simplicity and as a special example I shall consider the action of 

 Jupiter only, and also only his action upon those comets whose periodic 

 times are reduced to be less than Jupiter's period, the original orbits of 

 the comets being parabolic. In other words, how many of the 839 

 comets which, are reduced (Art. 35) to have periodic times less than 

 Jupiter's period will after perturbation have goals distant less than 15°, 

 30°, 45°, &c., severally from Jupiter's goal ? 



37. Let BA, fig. 19, be drawn to represent v^ and CA to represent 

 v,v^2. With A as a centre and AB and AC as radii describe the semi- 

 circumferences BLO and CHG. Let the angle BAH be made equal to 

 <u and BH be drawn ; then EA will represent the comet's velocity about 



