100] DEVELOPMENT OF r AND V IN SERIES. 169 
agreeing with the result in (53). Suppose k = 1; then the first 
equation of (57) gives 2a ( 2 2) = 1, whence u% = | sin 2 M. As an 
illustration involving both (57) and (58), suppose k = 1 and 
consider the second equations of (57) and (58). They become 
in this case 
'o ( x 3) + 3a ( 3 3) = 1, 
" < + 27 a? = £§ ; 
whence a ( p = — f, a ( 3 3} = + §, agreeing with the results given 
in (53). 
When the expansion is carried out by the method of Lagrange, 
or by that which has just been explained, the value of E to terms 
of the sixth order in e is found to be 
(59) 
E = M + e sin M + ~ sin 2 M 
+ 3^22 sin ~ 3 sin 
+ 4X2^ s ^ n — 4 • 2 3 sin 2 M) 
+ (5 4 sin 5M — 5 • 3 4 sin 3 M + 10 sin M) 
+ gy25 — 6 • 4 5 sin4il4 + 15 • 2 5 sin 2 M) 
+ 
100. The Development of r and v in Series. The value of r in 
terms of e and M can be obtained by the method of Lagrange by 
letting F(z) = cos E and making use of the last equation of (48). 
This method has the disadvantage of being rather laborious. 
It follows from Kepler’s equation that * 
dE _ e sin E 
^ de 1 — e cos E ’ 
dM — (1 — e cos E)dE. 
Therefore 
e dM = e sin E dE. 
de 
The method employed in this Art. is due to MacMillan.