496 Mr. H. Holt on a Method of Calculating 



Again, substituting the assumed values of p and v in equation (8), 



-7- 2p 2 -7- = — c 2 (4F 2 + 2c 2 G 2 ) sin c 2 t, 3m ; ^-gsin 2(v—v}) = Zk$mc l t' f 



hence 



-c 2 (4F 2 + 2c 2 G 2 )=0, or -4F 2 -2c 2 G 2 =0. 



Adding this last equation to the equation (A), we find 



fe 2 + P-l)F 2 +pB 2 = 0, or (c 2 2 +p-l)F 2 =~pB 2 . 

 Also 



c 2 G 2 = — 2F 2 . 



If we substitute for c 2 and B 2 their values as found by observa- 

 tion, viz. c 2 = ratio of motion of sun's anomaly to motion of 

 moon's mean longitude =-0748006, B 2 = 3e / --|e / 3 =-050286, 

 we shall have 



c£+ P-l= - -9916174, c 2 G 2 = - -00028372, 



F 2 = + -00014186, G 2 = - -0037930 ; 



or, reducing to seconds, 



Inequality in longitude = — 782"*36 sin c 2 t, 



Inequality in parallax = — 0"49 cos c 2 t. 



The coefficient of the annual equation thus found is not so 

 near the true value as that of the variation found in the last 

 article. It will be seen hereafter that the coefficient of sin c 2 t 

 receives some considerable corrections in the third approximation. 



Second Approximation. 



6. We have now calculated the approximate values of the two 

 chief inequalities of the moon's motion (the variation and the 

 annual equation), which are independent of the excentricity of 

 the orbit and its inclination to the ecliptic. If to these two 

 inequalities we add the principal term of the equation depend- 

 ing on the excentricity (which has for its argument the 

 moon's mean anomaly, and the coefficients of which must be 

 found by observation), the values of p and v will be of these 

 forms, 



p = 1 + F t cos cj + F 2 cos c 2 t + F 3 cos c 3 t, 



v = t + e + Gj sin c v t 4- G 2 sin c 2 t + G 3 sin c 3 t, 



c 8 t denoting the moon's mean anomaly, and F 3 , G 3 being the 

 coefficients of the principal terms in the equation of the moon's 

 centre. 



7. If these values of p and v be substituted in the differential 

 equations (7) and (8), there will arise a series of terms expound- 

 ing inequalities having for arguments the sum and difference of 



