:mi;moiks ok tiii; national acadf.my of sciexchs. 19 



tht'ir jtoriodic tiiiios roihaed below a given period will be e(|iial to the area inclosed in llie rone- 

 spondinj;- isergonal eiuve multiplied by the velocity perpendicular to tlie plane r„ sin 0, and by tlie 

 faetor J . U^w is the semiiuajor axis of the <>rl»it fur tlie limiting,' periodic time, tlie area of the 

 cories]>ondinji isergonal curvi' will be (article 17). 



iin^'a^ I -m'li cos rt mr x^ \ 



7T /im^'a^ / -m'l 



For (/S wc may, as before, take 2;r sin w dai, and wo shall then have 



■4»w2'«' / 2»»»'rt cos H 



^ ^T»^ r . ^. rivi^'a^ _ /2 m'ii cos h mr >,^~| 



dijj. 



The integration must extend through the iiositive values of the (piantity in srpiaic brackets 

 beginning at a) = 0. [In case r«) = gives a negative value for the ((uantity in square brackets we 

 must integrate between the two values of w corresponding to the zero value of the bracketed 

 ipiantity.J We may make s the independent variabh- by the ciinatioiis 

 mU = v'S sin&jrftt), I'uV'-i = XI', iiiid -.s cos 8= l—x'K 



These give: 



irrnmH frir^^-(^'^^''-:ll )'!,/«. 



33. If, now, we require the number of comets which, in each unit of time, shall i)ass the planet 

 in such way as that they shall have, after the passage, respectively, less thau one-half, once, three- 

 halves, and twice the planet's period of revolution 



we may place ® =rT*, and makeT equal, successively, to i, 1, f, and 2, andcom])utein each case 

 the value of X as given in the last article. The results are found to be Trnm^r-v multiplied sever 

 ally by the coefficients O-iaO, 0-92r), 1-875, and 2-943. 



31. By comparing tlic results of articles 27 and 33, and making the assumptions of article 

 20, we have the proposition that the number of cometn which, in a f/iren period of time, pii-sx their 

 perihelia nearer to the sun than a given planet, is to the number of comets whose perioilic times are re- 

 duced by the perturbing action of the planet so a« to be less severally than one-half, once, three halves, 

 and twice, the periodic time of the planet, as unity is to the square of the mass of the planet multiplied 

 severally by 0-139, 0-925, 1-87G, rtJirf 2-943. 



35. If Jupiter is the planet, w = i-gi;-^, and we may express these ratios as 

 1 ,000,000,000 : 12G : 839 : 1 701 : 2670. 

 That is, assuming the hypotheses of article 20, and regarding the planet as without dimension .so 

 as to intercept any coniets, if in a given period of time a thousand million comets come inparaboliv 

 orbitu 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 Jupiter ; 889 of them n-ill have their orbits changed into 

 ellipses with periodic time^ less than that of Jupiter ; 1,701 of them will have their orbits changed into 

 ellipses with periodic times less than once and a haif times that of Jupiter ; and 3,070 of them will 

 have their orbits changed into ellipses ivith periodic times less than twice that of Jupiter. 



30. Another and perhaps a more important inquiry is this. What etlect have the i)erturbatious 

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

 are reduced (article 35) to have periodic times less than Jupiter's period will, after perturbation, 

 have goals distant less than 15^, 30°, 45°, etc., severally from Jupiter's goal ? 



37. Let BA, Fig. 19, be drawn to represent r, and CA to r<>present v., ^2. With A as a center 

 and AB and AG as radii describe the .semicinuimlerences BLO and CIIG. Let the angle BAH be 

 made ecjual to co and BIl be drawn; then HA will reiuesent the comet's velocity about the sun, 

 BA the planet's velocity about the sun, and therefore MB the comet's velocity r„ in its orbit aTmut 

 the planet before perturbation. About B as center describe the semicircuiuference KIIT. Since 



