in the Secondary Systems. 497 



/3 denoting the periodical quantities arising from the disturbance. 



2M's . 2M's 



Since -^-3- is extremely minute, it may be replaced by - _ 3 , ~D l 



being the mean distance. The maximum and minimum values 



dr 2 

 of r may then be found by a quadratic on making -^ =0, and 



omitting the periodical terms denoted by /3. From the coefficient 

 of r in the resulting equation, it appears that 



M'_MV2D s'eUv\ 



D '-T r\m* + -y-)' ' • (33) 



M' 



-j- being the value of the mean distance in the absence of the 



. . 9AS 



disturbance, and st being equal to — -, the last expression 



o 



becomes 



D, = Constant =-5 — smW; . . . (34) 



and the secular diminution of J) l during each revolution will be 



5 -6548 As sin W . 



^ nearly, .... (35) 



in which z denotes the highest swell of the tides at the points of 

 the satellite in conjunction and in opposition with the primary. 

 The diminution which the disturbance occasions during the same 

 period in the relative eccentricity or the eccentricity divided by 

 J) l will be 



2-8274A*sinW 



Di « (36) 



These results may also be obtained by investigating the varia- 

 tion of the elements of the orbit according to the method of 

 Lagrange. If the tides could rise and fall on a satellite without 

 any impediments from friction, W would become equal to 180 

 degrees, and there could be no permanent change in the ellipse 

 which the body describes. It thus appears that the duration 

 of the secondary planets is much dependent on the absence of tides 

 from their surfaces; and perhaps the vast number of these 

 attendants belonging to the remote planets may be indebted for 

 their present existence to the intense cold, which keeps their 

 oceans in a perpetually frozen condition. 



Cincinnati, November 8, 1861. 



