7 2 Mr. Robertson on the Precession 
plane with the ellipse HKNG be HN, and let this ellipse be 
cut by the plane of the equator i the line KG. Then HN 
(14. XI.) is parallel to PL, and KG to EO ; and therefore 
(10. XI ) HN, KG cut one another at right angles. Again, 
as the axis PL is perpendicular to the plane of the equator, 
the plane of the equator is perpendicular (18. XI.) to the 
ellipse PELQ. The planes passing through the centre of the 
spheroid and the lines HN, KG are therefore perpendicular to 
the ellipse PELQ, and consequently (19. XI.) the straight line 
passing through the centre of the spheroid, and the point in 
which HN, KG cut one another, is perpendicular to HN, KG 
and also to PL, EQ. Consequently as the equator is a circle, 
KG (3. III.) is bisected by HN ; and as HN is parallel to PL 
it is a double ordinate to the diameter of the equator passing 
through the point in which HN, KG cut one another, and is 
therefore bisected in this point. Hence, as by a well known 
property of the spheroid, the ellipses PELQ, HKNG are 
similar, it is evident that KG is the transverse, and HN the 
conjugate axis of the ellipse HKNG. 
Let u — the distance of the centre of the ellipse HKNG 
from the centre of the speroid, and then as the points E, K, 
G, O are in the circumference of the equator, a straight line 
drawn from the centre of the spheroid to K or G is equal to 
a , and half of the straight line KG = V c?—u. Again, as the 
ellipses PELQ, HKNG are similar, a : e : : V a — if : -j\/ a* — if 
~ half of I IN ; and a : cf — if : : X : - ~ u xX = the area of 
a 
the ellipse HKNG. Hence, in order to find the disturbing 
force of the sun on the ellipse HKNG, instead of a* in the 
expression x — -x cf — e 2 * X we are to put cf—if, instead of 
