366 
NOTE ON THE LUNAR THEORY. 
[465 
where N, E, T, D, H, K, are arbitrary functions of c—g: P and Q denote given 
functional expressions. But, in order that r, v, y, considered as functions of c and g 
may be of the proper form, it is necessary as regards N to write simply N= 1 ; we 
have then 
r = elqr (E, c + D), 
v = l + K + P(E, r, c + D, c + H), 
y= Q(E, r, c + D, c + H), 
where E, T, D, H, K, are arbitrary functions of c — g; or, what is the same thing, 
writing for these quantities respectively e + Se, 7 + Sy, Sc, g — c + Sg, SI, where Se, Sy, 
Sc, Sg, 81 are arbitrary functions of c — g, we have 
r = elqr (e + Se, c + Sc), 
v =1 + 81 + P (e + Se, y 4- Sy, c + Sc, g + Sg), 
y= Q(e+ Se, y + Sy, c + 8c, g + Sg), 
that is, the values of r, v, y, are obtained from the elliptic values 
r = elqr (e, c), 
v=l + P(e, y, c, g), 
V= Q(e, y, c, g), 
by affecting each of the quantities e, y, c, g, l, with an inequality which is a function 
of c-g.