278 Prof. Challis on two new Theorems 



to mention, that, for the successful performance of most of these 

 experiments, considerable manipulative care is required. 



If, in the further prosecution of my inquiries in this direction, 

 I sliould discover anything worthy of notice, I hope to have the 

 pleasure of communicating it to you. 



I remain, dear Sir, 



Truly yours, 



Arthur H. Church. 



XLII. Proofs of two neio Theorems relating to the Moon's Orbit. 

 By Professor Challis*. 



THE theorems which I propose to demonstrate are con- 

 tained in a paper recently comnmnicated to the Cam- 

 bridge Philosophical Society. As I have reason to think that 

 they are new, and of considerable importance in the lunar theory, 

 I am desirous of giving them immediate publicity in the pages 

 of the Philosophical Magazine. They may be enunciated as 

 follows. 



Theorem I. All small quantities of the second order being 

 taken into account, the relation between the radius-vector and 

 the time in the moon's orbit is the same as that in an orbit 

 described by a body acted upon by a force tending to a fixed 

 centre. 



Theorem II. Tlie eccentricity of the moon's orbit is a function 

 of the ratio of her periodic time to the earth's periodic time, and 

 the first approximation to its value is that ratio divided by the 

 square root of 2. 



Let the moon's place be referred to the plane of the ecliptic 

 by rectangular coordinates, the origin being the earth's centre at 

 rest, and the axis of x passing through the first point of Aries : 

 let fjb= the sum of the attractions of the sun and moon at the 

 unit of distance, ?«'= the sun's attraction at the unit of distance, 

 and let r, r' be the radius-vectors of the moon and sun, and 6, & 

 their true longitudes, at the time t reckoned from a given epoch. 

 The following equations, including small quantities of the third 

 order, may be readily obtained : — 



^ = -(^-2p3J + ^eos(^-2^') 



^^y --(i^V- '"'y^ _ ^ sin (e-2&\ 



cPz __ /fju2 _fn'z\_ Bm'z 

 df-~\~^ 27V W^' 



* Communicated by the Author. 



