THE ORBITS OP THE EIGHT PRINCIPAL PLANETS. 39 



mean relative positions of their perihelia will always be the same ; and we shall 

 now inquire what their mean relative positions are. For this purpose we shall 

 resume equation (144:), and substitute in it the values corresponding to these two 

 planets. By this means we shall get the following equations, 



tan(cT' r -^-/i c )= 1 



^ /r sin Ky-gf^ +ff-fo l+jvy'sin l(flr 1 -flr )<+/3i-/3, l+&c. I (147) 

 N t ' r +N"coa j(0_^+£_ftj+JVTcos | (&-&)*+&-&!+&<!.' I 



tan(o r/ — g t— /?„)= 1 



iV"sin Kg-gJt+p-fcl+NS' sm \ (g l -g e )t+p 1 —p 9 \+&c. I (148) 



Now, since the mean values of the numerators of these equations are each equal 

 to nothing, and the signs of the denominators depend wholly on N ir and N 6 "; it 

 follows that cj' r will always be equal to er rj , if N e ir has the same sign as iV 8 "; and 

 OT ,r will always differ from of" by two right angles if N ir and jV u r/ have different 

 signs. According to the numbers which Ave have calculated, iV/ r and JV 6 ' V have dif- 

 ferent signs ; consequently, the mean longitudes of the perihelia of the orbits of 

 Jupiter and Uranus differ by a semicircumfercnce. 



For the purpose of determining the maximum values of tan (c/ r — g 6 t — /3 6 ) and 

 tan (c*™ — g a t — fi e ), we may suppose them to be of the following form, 



the coefficients of cos a being supposed positive. These equations evidently attain 

 their maximum values when cos a is equal to the quotient of its coefficient divided 

 by the constant term of the denominator, taken negatively. If we reduce them to 

 numbers, they will become 



0.0176673 sing 



tan (o —&<—&)— 0.0431601+0.0176673 cos a (15 ^ 



_ 0.0331038 sin a 



tan(ci —&<—&)— _ .0448614-|-0.0331038 cos a ( 15 ^) 



The first of these evidently attains a maximum value when a=±114° 10'; and 

 the second one when a=+42° 27'. Consequently, we shall find 

 Maximum value of (c/ r — g t— /3 6 )=±24° 10'; 

 and « » " (v v '—g 6 t-(3 )=180°±47° 33. 



The nearest approach of the perihelia of these two planets will therefore be 

 108° 17'. 



