16 ASTEROID SUPPLEMENT. 



PALLAS AND MERCURY. 



The values of a for Pallas and the principal Planets are so near the correspond- 

 ing ones for Ceres, that the coefficients for Pallas are readily obtained from those 

 for Ceres by means of the formulas and tables on pages 63 - 67. 



The inclination and eccentricity of Pallas are so great, however, that its pertur- 

 bations cannot be obtained to any great degree of approximation through the usual 

 development without a disproportionate amount of labor. The values of ¥'J and its 

 derivatives for Pallas will, therefore, probably never be needed for the computation 

 of its perturbations ; and especially for the action of Jupiter and Saturn. 



JUNO AND MERCURY. 



The inclination of Juno is 13°, and the eccentricity is 0.27. Both of these ele- 

 ments are large ; and the eccentricity especially is too great to render it necessary 

 to give the coefficients for this Planet. If, however, all or any of them shall ever 

 be needed, they may readily be found from the corresponding ones for Proserpine 

 given hereafter. 



To illustrate the use of the formulas for computing the variations of the co- 

 efficients corresponding to a change in the argument a, we will find the value of 

 log a 4 _D* ¥f for Juno and Jupiter from the value of the same coefficient for Proser- 

 pine and Jupiter. 



The formula is 



A log a 4 Dt b { ? = E 2 + (4 + 8 |3«j ^ . 



For Proserpine and Jupiter, 

 For Juno and Jupiter, 



a = 0.5104402. 



a = 0.5129176. 

 Sa= 2.4774. 

 With a = 0.5116790, the mean of the above values, which must be used when we 

 wish to take second differences into account, find from the Tables, pp. 25 and 67, 



Then, 

 Hence, 



log E. 2 = 2.0408, ^Af = 84.9, 2 /3 2 ^A? = 60.2. 



a a 



£ 2 Sa ='+272.1, ^A? Sa = 210.3, 2 /3 2 ^A? 8a = 149.1. 



A log a 4 D\ if = + 272.1 + 4 x 210.3 + 4 x 149.1 = 1709.7. 



From the general Tables we find for Proserpine and Jupiter, 



log a 4 Dt b ( f = 0.6459610, 

 For Juno and Jupiter, 



log a .D 4 b ( f = 0.6630584, 



a log a 4 D 4 bf = 1709.74. 



Products of 210.3 and 149.1 by 1, 2, 3, 4, &c. give the corrections corresponding 

 to these terms for all the coefficients. 



