12 A NEW METHOD OF DETERMINING 
node on the plane of the disturbing body. Let y be the angular distance from ascend- 
ing node of the plane of the disturbing body on the fundamental plane to the same 
point. 
If x, 7, are the longitudes of the perihelia, 
, 2’, the longitudes of the ascending nodes on the fundamental plane adopted, 
which is generally that of the ecliptic, we have . 
W=a—Q-—49, Woaw—Q—r. (3) 
The angles ®, y, 2 — 9’, are the sides of a spherical triangle, lying opposite the 
angles 7’, 180 — 2, J, - 
2, v, being the inclination of disturbed and disturbing body on the fundamental 
plane. 
The angles J, ®, 7, are found from the equations 
sin $ sin $ () + ©) = sin $(Q — 2) sin $(¢ + 7) 
sin 3 J cos} () + ®) = cos § (Q — Q’) sin 4 (¢ — 7) (4) 
cos $ Jsin $ (Y — ®) = sin $ (Q — 2) cos $ (¢ 4 72) 
cos § Lcos$ (J) — ®) = cos $ (Q — &) cos § (¢ — 7) 
In using these equations when Q is less than 9’ we must take 4 (860° + Q — 9’) 
instead of $ (2 — Q’). 
We have a check on the values of £ ®, J, by using the equations given in Han- 
SEN’s posthumous memoir, p. 276. 
Thus we have 
cos p. sin q = sin 2. cos (Q — &’) 
COS p. COS Y = Gos U 
cos p. sin 7 = cos 2. sin (3 — %) 
cos p. cos rT = cos (83 — &’) 
sin p = sin?’ sin (8 — 8) \ 6) 
sin J sin ® = sin p [ 
sin J cos ® = cos p. sin (? — 
(2 — q) 
sin sin (} — r) = sin p .cos (2 — q) 
sin J cos (Wy — r) = sin (7 ) 
( ) 
cos I = cos p. cos (@ 
