ON THE SPECIAL PROBLEMS OF DYNAMICS. 209 
98. In the case of a central force varying as (dist.)—2, the effect of a resist- 
ing medium (R O¢ v”) is considered in reference to the lunar theory, in the 
‘Mécanique Céleste,’ Book VII. ¢. 6. Formule for the variations of the 
elliptic elements are given in the ‘ Mécanique Analytique,’ t. ii. (2nd edition). 
But the variations of the elliptic elements are fully worked out by means of 
grec and Jacobian functions in Sohncke’s valuable memoir “Motus Corporum 
ve.” (1833). 
99. The effect of the resistance of the air on a pendulum has been elaborately 
considered by Poisson, Bessel, Stokes, and others; as the dimensions of the 
ball are attended to, the problem is in fact a hydrodynamical one. 
The effect on the spherical pendulum is considered in Hansen’s memoir 
Theorie der Pendelbewegung &c.” (1853). 
The effect on the motion of a projectile is considered in Poisson’s memoirs 
“Sur le Mouvement des Projectiles &c.” (1838). 
Liouville’s memoirs “ Sur quelques Cas particuliers ou les équations du 
mouvement dun point matériel peuvent s’intégrer” (1846-49). 
100. In the first memoir (§ 1) the author considers a point moving in a 
plane or on a given surface, where the principle of Vis Viva holds good (or say 
where there is a force-function U). The coordinates of the point, and the 
function U, may be expressed in terms of two variables a, 3, and it is assumed 
that these are such that 
ds*=)(da’ +d’), 
where ) isa function of a and 6. That is, we have T=3)(a!+4") ; and the 
equations of motion are 
d.dal_ 1dd- 0, am , dU 
di Oda” ae \t ay 
Ne deeee ot re 
ae Bap tO + ae 
One integral of these is 
A(a!?+B")=2U 40; 
and by means of it the equations take the form 
d.ral 1dr dU 
dt Papeete 
d.rp!_1 dro dU 
a ax gp CUtO+ 5, 
These equations, it is easy to show, may be integrated if 
(2U+C)\=fa—F@£, 
and they then in fact give 
Na?=fa—A, 
AN pP= A—FB, 
where A is an arbitrary constant. And we then have 
da dp 
Vja—A VA—FP 
which gives the path, and the expression for the time is easily obtained by 
means of a quadrature. 
It is not more general, but it is frequently convenient to employ instead of 
a, B, two variables » and y, such that 
a 2 2 
as ds*=)(mdp?+ndy’), 2 
