12 



MEMOUiS OF THE NATIONAL ACADIiMY OP SCIENCES. 



are tlieoretically two members of the faiscean. F()r points on tlie radical axis 'S =•/: and therefore 

 for this locus there is no change in the energy of the comet. 



17. Center and area of the isergonal ellipse. — The center of the isergonal ellipse is upon the 

 axis of h; making d=0, and solving for h we have 



h^ 



2m^ 



2m® / , 

 s sin i9\ 



•OS H- 



- \i 



.7^ )' ) 



2m/fly I I 



(14) 



The first term of the second member of (14) is the ordinate of tlie center, and the second term is the 

 semi-axis major of the ellipse. The ratio of the axes being 1 : sin ^, and As'^being=:»n»-, the area 

 ot the ellipse will be eqnal to 



47rw2®y 

 :.^sin e \ 



'i'asj J' 



LS. Maximum action of the planet. — For two particular values of ® the isergonal ellipses be- 

 come points. These values of ^c result if the maximum effect of the planet in increasing and 

 in decreasing the energy of the comet takes place, and they are obtained by making the two 



_f^=±l. Since at the same 

 2wa 



time /( =^2m'a>/.<i, we obtain 



7. A „.,-, ^ As 



Vi.Iues of /( equal to each other in (14); that is, by making cos ff- 



Let h' and /(", and 'ti ' and fe 

 construct the following tabh 

 ing planet: 



cos ff± V ^'"^ "^ =-2¥(co^^iri)- ^^-'^ 



be the positive and negative values of /( and T; in (1.")) and we may 

 )f their values. As in Table I, Jupiter is assumed to be the perturb 



•01443 

 •012.50 

 •00927 

 •00690 

 •00544 

 •00457 

 •00407 



•15174 

 •03307 

 •01290 

 •00654 

 •00387 

 •00253 

 •00179 

 •00134 

 ■00105 



2^85 

 2^69 

 2^61 

 2^61 



-10 •IS 



- 5^08 



- 3 13 



1^34 

 1^11 

 0^95 



•00426 

 •00489 

 •00598 

 •00789 

 •01149 

 •01934 

 •04192 



— ^00072 



— -00062 



— -00055 



— •OOOSO 



— ^00047 



— ^00044 



— ^00043 



— ^00043 



4^17 

 5^12 

 '6^60 

 9^09 

 13^71 

 23^70 

 52^36 

 206 ^30 



— 0^83 

 — 0-75 



— o^es 



—0 ^63 

 —0-60 

 — 0^57 

 — 55 

 —0 •54 

 — 54 



19. Explanation of Table II. — The meaning of the numbers in this table may be explained by 

 an example. If a comet moving in a parabola passes near to Jupiter, and the directions of the 

 two original motions at nearest points of the orbits make an angle of 10°, then the greatest ac- 

 tion of Jupiter (during the whole period of transit) in diminishing the velocity of the comet in its 

 orbit about the sun will take place if the two orbits actually intersect {d=0), aiul if the comet in 

 its unperturbed orbit arrives first at the point of intersection at the instant when Jujuter is dis- 

 tant therefrom •012.''>() (the earth's mean distance from the sun being unity). The resulting semi- 

 axis major of the comet's orbit about the sun will be 3-04. 



On the other hand, the greatest effect in increasing the velocity of the comet will take place 

 when the two orbits actually intersect, the comet in its uni^erturbed orbit reaches the point of 

 intersection later than the planet, and when the planet is distaut therefrom 0-15174. The semi- 

 transverse axis of the resulting kyin'ilxilic orbit about the sun will be 36-90. 



20. Resulting orbits of ma.vim ion pvrturlmlio)!. — The position of the relative orbit about Jupiter 

 in these cases of maximum perturbation for given values of (,) is easily determined. From the 

 equations (7), (6), and (1.5) 



tan <t=B/A=/i. sin rV/A=sin Hi (cos Ozkl). 



