ON THE SPECIAL PROBLEMS OF DYNAMICS. 193 
™ cos » 
sin ¥. ? 
a general symbol, the expansion being carried as far as e”, were given, Lever- 
rier, ‘ Annales de l’Observatoire de Paris,’ t. i. (1855). 
36. And starting from these I deduced the results given in my “Tables of 
the Developments, &c.” (1861); viz. these tables give (e=2-1), 
a 
| Ae wv"), 
Sete o 2 
Cae 51 saa oi") nh j=l to j=7, 
(Ce) (Je GY) 
(Ci) Go) Go) tain saa wat 
all carried to e”. 
37. The true anomaly f has been repeatedly calculated to a much greater 
extent, in particular by Schubert (Ast. Théorique, St. Pét. 1822), as far 
fraction. Such formule for the development of (Z —1 where 7 is 
. Yi . . : 
ase, The expression for — as far as e” is given in the same work, and that 
a 
for log - as far as e*° was calculated by Oriani, see Introd. to Delambre’s 
‘Tables du Soleil,’ Paris (1806). 
38. It may be remarked that when the motion of a body is referred to a 
plane which is not the plane of the elliptic orbit, then we have questions of 
development similar in some measure to those which regard the motion in the 
orbit ; if, for instance, z be the distance from node, ¢ the inclination, and a 
the reduced distance from node, then cosz=cos @ cos x, from which we may 
derive z=#-+ series of multiple sines of 2. And there are, moreover, the 
questions connected with the development of the reciprocal distance of two 
particles—say (a? + a'*—2aa! cos 0)~?—which present themselves in the pla- 
netary theory; but this last is a wide subject, which I do not here enter 
upon. I will, however, just refer to Hansen’s memoir, ‘‘ Ueber die Entwicke- 
lung der negativen und ungeraden Potenzen &c.” (1854). 
39. The question of the convergence of the series is treated in Laplace’s 
memoir of 1823, where he shows that in the series which express r and f in 
. . : . cos . oe 
multiple cosines or sines of g, the coefficient of a term sin 7 where 7 is very 
great, is at most equal in absolute value to a quantity of the form slg): 
A and X being finite quantities independent of 7, whence he concludes that, 
in order to the convergency of the series, the limiting value of the excentricity 
is e=X, the numerical value being e=0°66195. 
40. The following important theorem was established by Cauchy, as part 
of a theory of the convergence of series in general; viz. so long as e is less 
than 0:6627432, which is the least modulus of e for which the equations 
T : 
pues u, 1=ecos 
can be satisfied, the development of the true anomaly and other developments 
in the theory of elliptic motion will be convergent. This was first given in 
1862, 0 
