THE ORBITS OF 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(sr /r g 6 t /3 6 ) = 



-frl +&C.' 



tan(or rj -<7 < ,3 6 )= 



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 6 ' r and N 6 "; it 

 follows that cr /r will always be equal to cr", if JV 6 /r has the same sign as N 6 "; and 

 CT JF will always differ from o" by two right angles if N a ir and ^V 6 " have different 

 signs. According to the numbers which we have calculated, N 6 IV and N 6 yi have dif- 

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

 Jupiter and Uranus differ by a semicircumference. 



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

 tan (cs vl g 6 t /3 e ), we may suppose them to be of the following form, 



sin a. 



Bm a. 

 1 cos a' (l< 



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 



-&<-&)= -o6431601+6:oi76673 cos a 

 F/ _ _ 0.0331038^sina _ 



9tf Pe)~ _o.04486 f4^07)33T038 cos a' ^ ' 



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 (a'" g 6 t /3 6 )=24 10'; 

 and " " " (cr r/ gj (3 )=180+47 33'. 



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

 108 IT. 



