NOTE ON THE LUNAR THEORY. 
365 
465] 
we write m = 0 : it requires a little consideration to see that this is so, if in the 
coefficients only we write m = 0 ; but recollecting that c, g, stand for functions ct 4- C, 
gt + G, so that, for example, c — g, =(c — g) t + C — G, upon writing therein m — 0, 
becomes equal, not to zero, but to the constant value G — G, the identity subsists in 
regard to the coefficient of the sine or cosine of each separate argument ac + fig, 
and, consequently, it subsists notwithstanding that in the arguments c and g, instead 
of being each put = 1, are left indeterminate. And granting this (viz. that the 
equations are satisfied if in the coefficients only we write m — 0), then it is clear that, 
as above stated, the required values of r, v, y, satisfy the undisturbed equations of 
motion, if after the differentiations we write in the coefficients c = 1, g = 1. 
The required values of r, v, y, are of the form r = </> (c, g), y — ^ (c, g), v = l + % (c, g), 
but writing w = v + c — l, = c + x (c, g), the last mentioned property will equally subsist 
in regard to the functions r, w, y : in fact, v enters into the differential equations 
only through its differential coefficient —, and the differential coefficients of v and w, 
that is, of l + x( c > 9) an d c+%(c, g), differ only by the quantity c— 1, which becomes 
= 0, in virtue of the assumed relations c = 1, g = l. 
Hence the undisturbed equations are satisfied by the values r = </> (c, g), y = ÿ (c, g), 
w = c + % (c, g), when after the differentiations we write in the coefficients c = 1, g = 1 ; 
the foregoing values contain t through the quantities c, g, only ; and we have, therefore, 
d d d 
dt C dc + ^ 
d 
dg 
values r= </> (c, g), y = ^Jr (c, g), w = x (c, g), regarding r, v, y, as functions of c, g, 
satisfy the partial differential equations obtained from the undisturbed equations of 
cL cl cl 
motion by writing therein ^ -f ^ in place of . Hence also, considering r, w, y, as 
dg ' 
cl ci cl 
Hence, writing in the coefficients c=l, g = 1, we have ^ “ y~c ^ 
dc dg 
functions of c and c — g, then observing that ^ (c — g) is = 0, the values of 
r, v, y, satisfy the partial differential equations obtained by writing in place of ^ ; 
and inasmuch as these partial differential equations do not contain ^, they are to 
be integrated as ordinary differential equations in regard to c as the independent 
variable, the constants of integration being replaced by arbitrary functions of c — g. 
Consider the pure elliptic values of r, v, y, in an elliptic orbit with the following 
elements: A, the mean distance; A, the mean motion (A 2 A 3 = 1 and therefore A=N~*) ; 
E, the excentricity ; Nt+ D, the mean anomaly ; Nt + H, the mean distance from node ; 
Nt+K, the mean longitude; then writing c in place of t, we have 
r = A - s elqr (E, Ac + A), 
v (= l — c + w) = l — c + Ac + K + P (E, r, Ac + D, Ne + H), 
y = Q(E, T, Ac + A, Ac + A),