39. 



NOTE ON DR MORRISON'S PAPER (ON KEPLER'S PROBLEM). 



[From the Monthly Notices of the Royal Astronomical Society, Vol. XLIII. (1883).] 



THE reference to Hansen's paper should be made to Abhandlungen der 

 Siichsischen Gesellschaft dcr Wissenschaften, Band iv. p. 249, instead of to 

 Band n. as stated by Dr Morrison. 



In this paper Hansen's object is not merely to express the coefficients 

 of the series which gives the eccentric anomaly in powers of e, otherwise 

 this might have been done much more simply in the following manner. 



Calling g the mean, and x the eccentric anomaly, we have 



g x e sin x, 

 or x = g + e sin x, 



which is in the proper form for the application of Lagrange's theorem for 

 developing x or any function of x in terms of g and ascending powers of e. 



Hence we have 



x =g + e sin g + ^ (ain'flr) + ^-^ (sin'0) 



whence by substituting for the powers of sin<7 their expressions in sines 

 or cosines of multiples of g, and differentiating, we may readily obtain the 

 function of g which multiplies any given power of e. 



A. 38 



