42 RELATIONS PERTAINING SIMPLY [BOOK I. 



34 might converge too slowly ; and therefore it is by no means to be regarded 

 as a defect of the method about to be explained, that it is specially adapted 

 to those cases in which E or F has not yet increased beyond moderate values. 



* 



37. 



Let us resume in the elliptic motion the equation between the eccentric 

 anomaly and the time, 



where we suppose E to be expressed in parts of the radius. Henceforth, we 

 shall leave out the factor- \/ ( 1 -j-/u&amp;gt;) ; if a case should occur where it is worth 

 while to take it into account, the symbol t would not express the time itself after 

 perihelion, but this time multiplied by y/(l -j-fi). We designate in future by q the 

 perihelion distance, and in the place of E and sin E, we introduce the quantities 



EsmE, and E -^ (E sin E) = ^E+^ sin E: 



the careful reader will readily perceive from what follows, our reason for selecting 

 particularly these expressions. In this way our equation assumes the following 

 form : 



As long as E is regarded as a quantity of the first order, 



& E+ T V sin E= E J - E 3 + ^ E* etc. 

 will be a quantity of the first order, while 



E-s m E=\E* ^E 6 + - s faE etc., 



will be a quantity of the third order. Putting, therefore, 



6(ff sinJ?) _.. &E+^smE _ 

 -- 



will be a quantity of the second order, and 



^==l + dW^ 4 etc. 



will differ from unity by a quantity of the fourth order. But hence our equation 

 becomes 



