DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
311 
where u, v, are the variables conjugate respectively to f, and defined by the equations 
dT dT. 
U— -rit ^’=-777 > 
d^ d&' 
and W is T — U expressed in terms of §, 0, u, v. 
Now cos 0, y=^ sin 0, z = l ~\/ ; hence 
■ t(=1 
from which the following expression for W is easily obtained : 
('''•) 
Now let us take as a model for the solution of the above system, a set of normal 
integrals (in polar coordinates) of the system 
^ (B.) 
where In this system we have U= — ; and proceeding exactly as in 
art. 27, we obtain the following results : the two integrals of vis viva and of areas are 
c=v 
these are to be solved for u, v ; and then V is to be obtained from the equation 
dV =ud^-\-vd0. This gives 
V=c«+J*{2A-ny-i}*; 
and the remaining integrals are given by the equations 
dh 
— ^+ 7 , 
r and m being the elements conjugate respectively to h and c. Performing the dif- 
ferentiations and taking the integrals in the second term so as to vanish with 
the expression (see Appendix A.), we find easily the final equations 
n‘f-=.h-\-A^td — rfd.co^2n{t-{-r)^ 
I, (ii.) 
d§~‘^=h—\/ If—rfd.cos 2(0— tiy) j 
in which ^ is the angle between the axis of .r and a (distant) apse, and — r is the 
time of passage through that apse. The four equations (i.), (ii.) comprise a com- 
plete normal solution of the equations (B.). The last is the polar equation to the 
elliptic orbit ; and if we call a, h the semiaxes of the ellipse, we have 
c=nab, h=n^ ^ 
