( 739 ) 
where C represents a constant, dependent on the initial state. 
The first equation of (17) becomes by the introduction of 5: 
f =F"| R '' N ' h ' ^ ™ V .(19) 
By eliminating ip between (18) and (19) we arrive at: 
j^=====± ? m,Ji 1 , 1 V 4 '.4 
Let 
/q == p (i-£) - ( P :* + q ; + r)*, 
then f{Q ^> 0 for the initial value of but/Q < 0 for £ — 0 and 
^ = 1; so /(C) becomes zero for two values C, and C t lying between 
0 and 1. 
So £ will generally vary periodically between two limits. It may 
be expressed in the time with the aid of elliptic functions, after 
which $ lt x, and y are also known as functions of the time. 
For the extreme values zero and one of the modulus x of the 
elliptic functions (x — 1 / / // jj? —when the equation J{£) — 0 
has two real roots a and p besides and £,) we get special cases. 
Osculating curves. 
$ 17. At first approximation we have found: 
I /a 
where the «’s and slowly vary with the time. 
By introduction of C and tp and by change of the origin of time 
we find that we may determine the equation of an osculating curve 
by eliminating t between 
