516 
[219 
219. 
ON SOME FORMULAE RELATING TO THE VARIATION OF THE 
PLANE OF A PLANET’S ORBIT. 
[From the Monthly Notices of the Royal Astronomical Society, vol. xxi. (186J), pp. 43—47.] 
In Hansen’s Memoir, “ Auseinandersetzung einer zweckmässigen Methode zur 
Berechnung der absoluten Störungen der kleinen Planeten,” Abhand, der K. Sächs. Gesell. 
t. v. (1856), are contained, § 8, some very elegant formulae for taking account of the 
variation of the plane of the orbit. These, in fact, depend upon the following 
geometrical theorem, viz., if (in the figure) ABC is a spherical triangle; P, a point 
on the side AB; and PM, PN, the perpendiculars let fall from P on the other two 
sides AC, CB; then we have 
cos PM sin (BC + CM) = cos PN sin BN — tan \ C cos BC (sin PM + sin PN), 
cos PM cos (BC + CM) = cos PN cos BN + tan \ C sin BC (sin PM + sin PN). 
These equations, in fact, give 
cos PM sin CM — cos PN sin CN — tan \ C (sin PM + sin PN), 
cos PM cos CM = cos PN cos CN;