MK.MOIUS or TIIK NATIONAL ACADKMV OK aciKNCKS. 



21 



Ati-.uu l.y tl 



L.-t </-' ami </' 



si'iitiiiir the V 



.!!>. If(.)" 



fqiial, hence 





■» ^ .■J-«'— 2 cos' ft)"i 2 cos w" («' 

 r " !t-8cos' u,"—(\h- + s 



' ho tlie tw.. values of </« <.l)taiiie(l by aiibstitiitii 



ihie f..i- the jxiiiit nearer to A. 



= !Kto, ji,,,! tlierel'ore cos (i>"=0, we liavc aloui;- II 



-sin" m")' 



/allies of -; 



1 , ,, 8'-l 



.^^, and </"=^^ 



tluit the iiuml 

 V 



4 



)f comets iiaviiifj <|iiits less than 'M)° from Jiipiti 

 nnrm-) 



;fr'j*-r'[(»-:-l5-;: 



|iiil and ''«<»• is 



(V+v/1 



= •7(111' ;7Hrm'>-^ 



Siiiee the whole miiuber of such comets is (Art. 'S.i) equal to •!t2.'» Tnirin^r', the number of comets 

 the distance of whose quits from Jupiter's quit is between 1)0° and 180° is -224 nntm^r^. The 

 number of the comets for which <« < r that have inclinations to the ecliptic less than 90^ is to the 

 number that have inclinations greater than 90^ as 701 is to 224. Of the 839 comets njxilrii of in 

 article SH, 263 will after prrturhntion hare retroijraile motions, and 030 will hare direct motions. 



40. If &?" is less than OO^ the expression to be integrated in order to cover the area SAKU 

 will be 



sponding expressi< 

 I ^„ds — I <?" ds. 



If at" is greater than !H)° the corresponding expression becomes 



As the value of® introduces into ^' ami ^" only one radical in s, and that a radical of the second 

 degree,these integrations are possible. Fiirite summation is, however, more convenient. Comput- 

 ing the values for each interval of \'>°, weeonstrnct the following table. The first column indicates 

 the interval in values of co"; the se<;ond column gives that coefficient of \7rnrm' r- t\\;\t must be 

 used to obtain the number of comets, which in a unit of time will pass perihelion nearer than 

 Jupiter's distance to the sun, shall also have their periodic times reduced to be less than Jui)iter's 

 period, and shall also leave Jui)iter's vicinity so that the distance between the quits of the two 

 bodies is between the two values in column I; the third column indicates the distribution of the 

 839 comets of article 3(5 through the twelve zones. 



- Tablk III. 



