T1IE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 37 



If these values be substituted in equations (C), we shall have the complete values 

 of h, h', h", &c., /, l\ I", &c., from which we can obtain the eccentricity and place 

 of the perihelia by means of the formulas, 



tan or =7t-f-Z ; e=-j/7t 2 +Z z =7j cosec c*=Z sec or. (139) 



19. If, in equations (C), we put, instead of Ji and Z, their values, e sin j, and 

 e cos or, we shall get 



e sin a=N sin (</<+/3)+ JV; sin gJ+pJ+N, sin (#,*+&)+ &c. (140) 

 e cos s= JV cos (<7<+/3)+AT' oos fcH-jffi)-j--ft| cos (Jtf+ftH- &c - ( 141 ) 



Multiplying equation (140) by cos (gtf+/3)> and (141) by sin (</<+/3), the differ- 

 ence of their products will give 

 e sin (s-j7-/8)=^i sin | (fc-^+ft-/? } +iV 2 sin { (g t -g)t+p a -p \ +&c. (142) 



If we multiply equation (140) by sin (gt-\-(3), and (141) by cos (#<+/3) the sum 

 of their products will give 



e cos (a gt /3)= 1 M43} 



JV+JV- 1 cosl( i7l -^+/3 1 - / 31+^cos|( fl r 2 -^+ l 8 2 - / 3|+&c. j k 



Dividing equation (142) by (143), we shall eliminate e, and get, 



^ sin | (ft-^+ft-ff | + JV 2 sin 1 (g a -g)<+ft-ff | +& c 



a(a-flrt-^-JV + JVTCOS j fa-gy+pt-p | + Aicos 1 (gt-gyt+fr-p \ +* **' 



When the sum JV r 1 -(-^ r 2 -}--^ r 3+&c., of the coefficients of the cosines of the denomi- 

 nator, all taken positively, is less than N, tan (or j7< /3) cannot become iniinite ; 

 the angle is gt (3 cannot therefore become equal to a right angle; consequently, 

 the mean motion of the perihelion will be in that case equal to gt. It is also easy 

 to see that when all the cosines of the denominator of tan (EJ gt /3) become equal 

 to 1, N lt N 2 , N s , fec., being supposed positive, the denominator will be either a 

 maximum or a minimum, and the numerator will be equal to nothing ; tan (tsgt 

 /3) will then be equal to nothing; or gt /? will therefore become equal to 

 nothing, and equation (143) will become, 



e=N I #i+JV 2 + JV 3 +&c. I ; (145) 



Consequently, the maximum and minimum values of e will be 



maximum e=N-{- N l -\-N 2 -\-N 3 -\-&c.; ) fid.R-\ 



and minimum 6=N\N l -\-N 2 -\-N 3 -\-8cc.\ J 



We shall now substitute the numbers which we have already computed, in these 

 equations, for the purpose of determining the maximum and minimum values of 

 the eccentricities, and the mean motions of the perihelia. 



20. For the planet Mercury, we have, 



Maximum 6=^+^+^4-^3+^+^+^+^=0,2317185. One-half of this 

 is 0.1158593, which being less than N, it follows that N exceeds the sum of all the 

 remaining terms ; consequently, the mean annual motion of Mercury's perihelion 

 is equal to g or 5".4638027, and performs a complete revolution in 237197 years; 

 and the minimum value of the eccentricity is 0.1214943. 



