and on the Rotation of a Heavenly Body. 209 



Thus we find 



log M, = l -273001 3, logH; =4-7178693, 



log M2= 1-1200339, logwg =4-1762574, 



logW3=2-7275880, log?/y =5-6373125, 



log W4=2-2648165, log «,„= 5-0723701, 



log W5= 3- 7675362, logM,i = 6-5128775, 



log M6= 3-2495999, log Mi2= 7-9495712. 



Hence we have 



So that the supposition a = i is too small. The above numbers 

 may be conveniently employed in calculating the values of m'i, 

 w'2, &c. corresponding to any other value of a.. For 



logM'„=logM„ + 2nlog (2a). 

 Suppose «= 4 ="625 ; 



.-. logM'„=logM„ + nx -19382; 

 and we shall find, as before, 



Therefore the supposition a = -625 is too great. 

 Suppose «= -621, 



.-. logw'„=logM„+rax -1882432. 

 We shall find in the same way, 



1_1 .sQy (-621)2-3 .5(i-^)'(-621)4-&c.= -0025. 



Hence the true value of — lies between -621 and -625. By the 



a 



rule of double position we find 



a! 



— = •6213, 

 a 



which is very near the true value. If the excentricities of the 

 two orbits be taken into account, we shall have 



o da 



+ a'e^ + /3'eV + y'ee''^ + S'e'^ + &c., 

 where a, /9, &c. arc functions of a, d, &c. 



If c, ff be regarded as independent variables, we have 

 = 2«e + ySe' + Sa'e^ + 2/3'<?e' + ^e'^ + &c., 

 = 27e' + /9e + SSV* + 27'ee' + ^'e'^ + &c. 

 Phil. Mag. S. 4. Vol. 15. No. 99. March 1858. P 



