(545) 



THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 169 



Therefore Ave shall have 



p =Nsm (^-L.0-_ft)+jy i sin (&<+ft— ft)+iVi sin (g^+ft-ft) " 

 +iV 3 sin ( fl r 3 <+ft_ft)+iV 6 sin (</ 5 *+ft-ft) 

 +iV sin (^+/^6— /3 4 )+iV 7 sin (</ 7 <+ft— ft) ; 



?0 =iy r cos(^+/3-/3 4 )+^ 1 cos (^+ft— ft^^cos^+ft-ft) 

 +iV 3 cos (^+ft_ft)+ iV 6 cos {g b t+p b — ft) 

 +iY cos (^+ft_ft)+ TV, cos (<7 7 <+ft-ft) 5 . 



jp '=^ sin (^+/?_ft)+A7sin (fc*+ft— ft)+iV 3 ' sin (^4-ft-ft) 

 +JVr'sin(^+ft-ft)+&c.; 



5 '=i^' cos (^+/3— ft)+i\Y cos (&*+&— &)+tfs c° s (fltf+ft— ft) 



+iV 3 ' cos Q/^+ft-ft) +&c. 



(546) 



Substituting for jp and </„ their values given by the first members of equations 



(537), we shall easily find 



sin <p sin (0 - (J t-(3)=N 1 sin { (^-gty+ft—/? } +iV 2 sin j (^—^-fft—/? 1 ^ 



+iV 3 sin j^-^+ft-^+i^sin j (5 r 5 -^+ft-/?j I (547) 

 +iV sin { (i7o— Sr)<+iS d — ^ } + JV 7 sin j(,/ 7 _^+ft-/3j J 



sin <p cos (0 O — <7< — (3) 



=N+N y COS f (0,-^+ft-/? | +iV 2 COS { (g a -g)t+p 2 -p \ \ 



+iV 3 COS j (g 3 -g)t+p,-P \ +iV 5 COS J (gr.-^+ft-/? j f ^*°J 



+iV 6 cos | (g 6 -g)t+p 6 -p \ +iV 7 cos { f^-^+ft-^ j j 



From these equations it is easy to show that the mean motion of O is equal to 

 gt when N exceeds the sum of the coefficients of the cosines, all taken positively. 

 We shall also have 



maximum <p =N+ N^M+Nz+Ns+N^Nt; ) ,^g. 

 and minimum. <p =N— (iVi+i^+^+^+iVg+iV;). j 



4. If we now substitute in these equations the values given in Chapter II, § 7, 

 we shall obtain the following maxima, minima, and mean motions. 





Inclination to invariable pis 



me. 



Mean motion of 

 nodes in a 

















Maximum. 



Minimum. 



Julian year. 



Mercury, 



9° 



10' 



41" 



4° 44' 



27" 



—5". 126076 



Venus, 



3 



16 



18 











indeterminate 



The Earth, 



3 



6 















tc 



Mars, 



5 



56 



2 











u 



Jupiter, 







28 



56 



14 



23 



— 25".934567 



Saturn, 



1 







39 



47 



16 



—25 .934567 



Uranus, 



1 



7 



10 



54 



25 



— 2.916082 



Neptune, 







47 



21 



33 



43 



— 0.661666 



22 March, 1872. 



