47 



gressive motion of the perigee thus found is, as is well known, 

 only about half its real motion. 



We must next investigate the effect of the new substitution 

 u= 1- — cos (i*— t) + — cos (»— m f + t) on the terms of the 



equation (c) of art. (1.) and thence deduce a new equation (d) the 

 integration of which will give a new value of u. For this 

 purpose it will only be necessary to compute the variation of 

 equation (d) arising from a variation 5 m = -1^ cos (i/— m d + tt) 



and as the object of our enquiry is the mean motion of the Lunar 

 perigee, it will be only necessary to compute the new terms of the 

 equation (rf) of the form B cos (v — tt), for from a similar term, viz. 

 _ ^!^ cos (i-— tt) arose the term cos( (1 — iJ^^y—A. The inte- 

 gration given in case 3 of the preceding article will then at once 

 give the coeflScient from which the mean motion of the perigee 

 is deduced, as far as the new approximation to the value of u is 

 concerned. 



To compute the variation of the equation (<f) as to the above- 

 mentioned terms. 



1. i- -Tin = — ir-3-r-4 — tm—, cos (2 *— 2 ") -i — rr—- sm (2 »— 2 f) 



t 



II = m V — 2 e m sin (n — w) 



This gives 3v r: — 2 e m sin (v — tt), supposing 5 m =: — cos [v — n) 



Hence nearly I v = — 2 ^ e m sin {v — m v + -n) 

 when 3 M = — cos (v — 2 m n + t) 

 Substituting these values of 3 m and i f, 



