272 SEE— SECULAR EFFECTS OF THE [April 21, 



Here )i is the mean motion and E the eccentric anomaly. It will 

 be seen from the first of equations (6) that the semi-axis major is 

 diminished by a secular term depending on t, and by a periodic term 

 depending on the difference of the sines of the angles E and -Eg, or 

 the position in the orbit. Thus the mean distance is subjected to 

 both periodic and secular variation. 



In the case of the eccentricity, however, the second of the equa- 

 tions (6) show's that there is no secular term, and only periodic 

 changes occur. A similar remark applies to the longitude of the 

 perihelion as shown by the third equation -of (6). 



We conclude, therefore, from Stromgren's careful analysis that 

 there is no secular decrease in the eccentricity due to a steady 

 growth of the central mass; and that the views expressed by Peirce 

 and Bryant are due to confusion, or to some error in the chain of 

 reasoning. 



This conclusion accords with the result reached by Professor 

 Lehmann-Filhes, in paper Xo. 3,'' cited above. For Lehmann- 

 Filhes shows that 



c cos TT = r„ cos TT,, + periodic terms, 1 

 e sin TT^ (',, sin tt,, -|- periodic terms : J 



and remarks that when the attracting mass slowly increases the 

 orbit slowly narrows up, but yet always remains a similar conic 

 section. He adds that this is true for any eccentricity whatever. 

 The results of Lehmann-Filhes and Stromgren, each worked out 

 independently of the other, and with much detail, are therefore in 

 entire accord; and as Str(">mgren's development is given in full, and 

 every step in liis analysis is quite clear, we must reject the conclu- 

 sions of Peirce and Bryant as not v.'ell founded. 



This concllusion that the steady increase of the central mass will 

 not diminish the eccentricity also confirms the results reached by 

 Airy* and by Sir John Herschel.-^' For these eminent authorities 

 show that a central attractive disturbance decreases the eccentricity 

 as the planet moves from the perihelion to the aphelion, but increases 



^Cf. A. N., 3479-3480. 



* ■' Gravitation," pp. 50-51. 



'"Outlines of Astronomy," tentli edition, 1869, p. 463. 



