ON THE SPECIAL PROBLEMS OF DYNAMICS. 191 
upon that of a force P, so that the motions for the forces Arts and B +5 
y* 
are deducible from those for the forces Ar and 5 respectively. Some other 
laws of force, ¢. g. S+Br, Stat o+e, are considered by Legendre, 
«Théorie des Fonctions Elliptiques” (1825), being such as lead to results 
expressible by elliptic integrals, and also the law Ll for which the result in- 
r 
volves a peculiar logarithmic integral. But the most elaborate examination 
of the different cases in which the solution can be worked out by elliptic 
integrals or otherwise is given in Stader’s memoir “De Orbitis dc.” (1852), 
- where the investigation is conducted by means of the formule which give 
¢ and @ in terms of r (ante, No. 14). 
26. In speaking of a central force, it is for the most part implied that the 
force is a function of the distance: for some problems in which this is not 
the case, see post, Miscellaneous Problems, Nos. 86 and 87. 
It is to be noticed that, although the problem of central forces may be (as 
it has so far been) considered as a problem in plano (viz. the plane of the 
motion has been made the plane of reference), yet that it is also interesting to 
consider it as a problem in space; in fact, in this case the integrals, though 
of course involved in those which belong to the plane problem, present them- 
selves under very distinct forms, and afford interesting applications of the 
theory of canonical integrals, the derivation of the successive integrals by 
Poisson’s method, and of other general dynamical theories. Moreover, in 
the lunar and planetary theories, the problem must of necessity be so treated. 
Without going into any details on this point, I will refer to Bertrand’s 
memoir “Sur les équations différentielles de la Mécanique ” (1852), Donkin’s 
memoir “On a Class of Differential Equations &c.” (1855), and Jacobi’s pos- 
thumous memoir, “ Nova Methodus &c.” (1862). 
Elliptic Motion, Article Nos. 27-40. 
27, The question of the development of the true anomaly in terms of the 
Mean anomaly (Kepler’s problem), and of the other developments which pre- 
sent themselves in the theory of elliptic motion, is one that has very much 
occupied the attention of geometers. The formule on which it depends are 
mentioned anté, No. 15; they involve as an auxiliary quantity the excentric 
anomaly wu. 
28, Consider first the equation 
g=u—esin u, 
‘which connects the mean anomaly g with the excentric anomaly wu. 
_ Any function of u, and in particular wu itself, and the functions ee nu may 
be expanded in terms of g by means of Lagrange’s theorem (Lagrange, ‘ Mém. 
de Berlin,’ 1768-1769, «Théorie des Fonctions,” c. 16, and “Traité de la 
Résolution des équations Numériques,” Note LE): 
29. Considering next the equation 
tan af=/ tie tan 2 u, 
which gives the true anomaly in terms of the excentric anomaly, then, by 
/ replacing the circular functions by their exponential values (a process em- 
