42 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
The notation employed for the disturbed body is 
a 
t = longitude of perihelion ; 
= longitude of the ascending node; 
= inclination to the ecliptic ; 
= mean distance from the sun; 
= mean annual motion ; 
= eccentricity ; 
eccentric angle = arc sin ¢; 
= radius vector; 
= true anomaly ; 
= eccentric anomaly ; 
m = mass; and the same symbols with accents, =’, 6’, 7’, etc, rep- 
resent corresponding quantities for the disturbing body. 
From the definition of the secular perturbations, according to 
Gauss, the perturbing function may be expressed by 
22 2x 
m’ 1 — e cose E 
V= fa f fA ae de 
0 9O 
in which p is expressed by 
»%K,24 0.3 2 = 
I 
p? =a"? +r — 2dr cos (a, 7). 
We also have 
r =a’ (1— ecose)’ 
a'r cos (a’,r) = a’ cose” {cos m (a cos ¢ — ae) — a cos ¢ Sin « sin r} 
+ a’sin < {a cos g sin e cosz + sinz(a cose — ae)}. 
Putting for brevity 
a’ (1 — ecose)? = pP, 
2 {cos z (a cose — ae) — a cos gsin esin z} = 4, 
2 {sin m(a@ cos ¢ — ae) + @ Cos ¢ SiN € Cos z} pa 
we get 
ey Ai) ae ’ A 
p= (a” + p.) — a (g, cose’ + 8 sin ’). 
The differential equation for the secular variation of the peri- 
helion is 
