220 
MR. IVORY ON THE THEORY 
Let ixr — -^3, n being the constant of the mean motion in the primitive ellipse, 
when t = 0 : then 
ndt = d{(l + 2 &f d -^d£f "I 
y . . . (13) 
dZ, = ndt-di{z a .f i 1 ^d^+ |a2(y^|rf|) 2 &c. I 
Taking next the semi-parameter h 2 , we have, by equation ( 3 ), 
hdh = r*(d'R-^dr): 
but d'R = (^) dv+ (^) dr ; wherefore, 
hdh = G?) -^ = . (“) <»«. 
In order to find the value of (^~)? let the expressions of ^and^ in the 
d v d v 
formulas (C), be added : then, since it has been shown that + -j— = 1, we 
get, 
JR _ /JJRA (dr Jr\ (dR\ . 
'dm \ d r ) \d m' dv) * \dv ) 
dR . dR 
d 
d t d t 
and, because r is a function of v — w, = 0 ; wherefore. 
_i_ ^ R _ 
'dm \ d v / 
JR . JR 
de 
Further, because g always accompanies £, or which is the same thing, because 
R is a function of £ + g, we have ^ ; 
consequently, 
JR , JR /JR\ 
J £ "“"Jct \ Jd / ' 
By substituting this value, 
hdh = #JT=7‘.(^ + i £)di, 
■v = a (i - A) + a/>yr=F(af + ■£) 
( 14 ) 
