MK.MOIKS ()|- rill': NAI'IONAl, .\( Al>i;\n Ol' S( II'.NCKS. 



Vi 



The positive sigu fjives '2i>= ', iiiid tlio mi;;ativt> si^jii j;jve.s U(r=T-(-''. I5iil fUi- iui^li- 2(r in 

 the lirst i-ase ia the iiii','hM>t'tlie iwymplntes inehisiiifj the hraiich of the hyper))ola desciilM-d about 

 .Iiipiter by the (tmiet. Since the two oriffinal orbits intersect, the phme oC the rehilive oil>it 

 contains the planet's patli, so that the conu-t jiasscs directly in front of the planet and beiiitr 

 tnrned backward leaves . I iipitei- exactly in Mie direction of .Inpitei's (put. • Tin- jilace ot enconnter 

 with Jupiter will be near an apse of tlie comet's resultint; orbit about the sun. The comet leaves 

 the planet with the relative velocity (>(_,, so that if \<1 the nu>tion about the sun in the new orbit 

 will be direct; if «>1 the motion in the new orbit will be retrofjjrade. Tliat is, by (4) when 

 . < JT the resulting motion is direct; when (.'>.^?r the resultinj; motion is retrofjrade. 



In the second case the angle '2ir, being greater than l.SO \ stainls for the angh- between the 

 asynii)totes exterior to the orbit. Hence the comet passing beliind the planet will be turned 

 I'.irward and will leave the planet in the direction of Jupiter's goal, and have a velocity that will 

 send it permanently out of the solar system. 



•Jl. The results of article 20 assume that u> is given. To liinl f-.r what value of <.» the period 

 of the resulting orbit is the shortest possible we may put Ax'=mr and 1 — «^=2« cos rt in (15) so 

 that 



This gives .s 

 But when «= 



■ ± 1, in which result 



To find the minimum for </ place , =(' m ihi^ ctpi.! 



since .V is inherently positive only the posit i\e sign is used. But when s—i,'a=^r,h=mr and 

 a =477-. Ucncc the f/rcdfeiit effect of perturbation of a ptumt moving in a circular orbit in shortening 

 the periodic time of a comet orii/inally moriny in a parabola is obtained if the coniefs original orbit 

 actually intersects the planefs orbit at an angle of 15^, and if the comet is due Jirst at the point of 

 intersection at the instant irhen the planet's distance therefrom is equal to the planefs distance from 

 the sun multiplied by the ratio <f the mass of the planet to the mass of the sun. 



Fig. 3; ta^nO" 



The relative velocity of the comet on leaving the planet's sphere of action wonld be equal to 

 and directly ojiposite to the planet's velocity fx=l), and the comet w<mld be left entirely at rest to 

 fall to the sun. This case could not hai>pen fur lilaiicts like the earth where mr is less than the 

 seinidiaineter of the jdaiiet. In the case of the earth mr is less than 300 miles, and actual collision 

 would result. But for Ju])iter mr is greater than the distance of the second satellite from the 

 ]»lanet. The nearest ajtin-oach of the comet to the planet wonld be mr( •/'_' — 1), which is more than 

 four times the radius of Jujiiter. Ileiiee, this case of maximnin diminution of major axis could 

 occur near Jupiter. 



L'L*. Tsergonal ellipses for u)=lin.—ir \vr i\\:\kv u'=\{n the vanishing points of the isergonal 

 ellii>ses will be (Table 11) at rf=0, /(=:-0I2r.(). and </ = (», /( = --1.517l. In Fig. 2 let OH and OH be 



filial anil tlio '/u(( of 

 I'lo-ly 18 miiving. 



iiovint; body 1 



