July 25, 1902.] 



SCIENCE. 



133 



finite, determinable intei'val of time. The 

 equations to be integrated are of the type 



(1) ~ = X,i.v„-,x„;a„-,aj-l3„-,P>>;t) 

 (1 = 1, -, «), 



where the x,. are any variables defining the 

 position and motion of the moon, and the 

 a. 's and 5 's are parameters occurring in the 

 differential equations. 



Solutions as power series in the « 's are 

 sought of the form 



(2) 



where the 



(i = l, -, n). 



are functions of the time to be determined. 

 Substituting (2) in (1) and equating to 

 zero the coefficients of the various powers 

 of the o 's, it is found that after the 



have been found the other coefficients are 

 determined by linear non-homogeneous 

 differential equations which can always be 

 solved. The proof of the convergence of 

 these series is made by employing suitable 

 comparison differential equations. 



There is nothing to prevent any of the 

 /3's being numerically equal to any of the 

 a 's. In fact, on the start all the parameters 

 are a's, but before the integrations made 

 those which occur in a special manner, as 

 in the trigonometrical functions, may be 

 called /?'s. "When this is done in an appro- 

 priate manner all the 



are purely periodic functions of the time 

 except the angular variables, each of which 

 has one term which is proportional to the 

 time. After a finite number of terms have 

 been found they may be rearranged as 

 Fourier series whose coefficients are power 

 series in the a's, giving expressions of the 



same form as usually found by lunar 

 theorists. 



The advantages of this method are: (a) 

 The series are known to converge, ( b ) every 

 step is defined in advance and contains 

 nothing arbitrary, and (c) the work is 

 divided up in a convenient manner. 



The Mass of the Rings of Saturn: Professor 



A. Hall, South Norfolk, Conn. 



The mass of these rings was first deter- 

 mined by Bessel in 1831 from the motion 

 of the apsides of the orbit of Titan. This 

 motion is about half a degree in a year. But 

 the action of the figure of the planet, and 

 the attractions of the other satellites were 

 neglected ; and, as Bessel pointed out, the 

 resulting mass of the rings was too great. 

 This mass is 1/118, the mass of Saturn 

 being taken as the unit. 



In this paper an equatioii was formed 

 containing two indeterminate quantities 

 depending on the figure of the planet, the 

 mass of the rings, and the masses of the 

 three brighter satellites, Rhea, Dione and 

 Tethys. The small resulting action of the 

 other satellites was estimated. The coeffi- 

 ciency of these six indeterminate quantities 

 can be computed with sufficient accuracy. 

 The uncertainty in finding the mass of the 

 rings arises chiefly from the lack of good 

 values of the masses of the satellites. 

 These masses must be found from the mu- 

 tual perturbation of the satellites. Substi- 

 tiiting the values of the masses of the sat- 

 ellites determined by Professor H. Struve, 

 the principal coefficient depending on the 

 figure of the planet was assumed to be 

 0.0222. The mass of the rings is 1/7092. 

 It is probable that Struve 's masses of the 

 satellites are too small, and the above mass 

 of the rings too great. 



Saturn will soon return to our northern 

 skies, and it is hoped that further observa- 

 tions and their dimeiisions will give good 

 values of the constants of this interesting- 

 system. 



