212] A MEMOIR ON THE PROBLEM OF DISTURBED ELLIPTIC MOTION. 
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equivalent, of course, to two equations. The first and second of them give in fact 
x, y, in terms of z and (f>. 
The value of z is 
*=C+/, 
so that x and y are given functions, and the longitude v is given in terms of x and 
6 by the equation 
v = x+ d, 
and consequently the three coordinates r, v, y, are by the system of equations given 
in terms of t and the elements. 
From the equations which connect z, x, y, cf>, treating all these quantities as 
variable we deduce 
sec 2 x dx = cos 0 sec 2 zdz — tan z sin cf> d(f>, 
cos y dy = sin </> cos zdz + sin z cos </> cZ</>, 
sec 2 y dy = tan </> cos xdx + sin x sec 0 dcf), 
sin zdz = cos y sin xdx + cos xsin y dy, 
equivalent of course to two equations; and the system is easily reduced to the more 
convenient form 
dx = cos <f> sec 2 y dz — tan z cos 2 x sin d(f>, 
dy = sin </> cos xdz + cos x tan z cos </> d(f>, 
dx = cot cf) sec x sec 2 y dy — tan z cosec <£ d<p, 
dz = cos (f) dx + cos x sin <£ d(j>, 
joining to these equations the 
dz = dtj + df, 
dv = dx + d6, 
and considering at present the mere analytical forms, first if dcf> =0, dt> = 0, we have 
dx = cos 0 sec 2 y dz, 
dy = sin </> cos x dz, 
dz = df, 
dv = dx. 
Next, if dy = 0, dv = 0, we have 
dx — — tan z cosec cf) d<f>, 
dz = — tan z cot <£ dcf), 
dz — cos (j) dx, 
dx — — dd, 
dz + cos cf) dd = 0. 
I remark also that the equation, tan y = sin x tan <f>, may be written in the form 
cos 2 (j) sec 2 y + sin 2 (f) cos 2 x = l. 
C. III. 
35