Bowditch'S HemarTcs on JDoclor Stewavt^s Formula, 115 



+ 



Substituting this ia the equation (1), which may he put under 

 a form j7 = i- m^ . \2 — 20 A,^'' 4. 10 (i + m) . A, ^^ } it becomes 



p ^i.m'li + y m+'^' m~\ (6) 



Now the mean motion of the moon from apsis to apsis being 



360° . (1 — -i^H = 360° (1 + i.;? + l/j* + §*c.) the motion of 



360° 



c 



the apsides in the same time will be 360° (^^ -{- ^ p* + Sac), and 

 by using the preceding value ot p it becomes* 



-|m\ Jl + V w + ^-|4^ w^ ! .360° (7) 



If we substitute the value of m = 0.0748013, corresponding to 

 to the moon, the three terms of this formula will become respec- 

 tively 5439^ 3814^^, 1313^, whose sum is 10566^^ = 2°o6' 6^, 



which differs a few 



4/74-''' on account 



I 



of the neglected terms. The first of these numbers 5439^ is 

 nearly equal to the sum of the two last 51^7^^, so that the mo- 



V 



tion is nearly doubled by noticing these two terms. This was 

 not done fully by D'Alembert, Euler and Clairaut in their first 

 approximations towards a correct lunar theory, so that they then 



made the motion only half its real value.f The term A^^'^ seemed 



* The two first terms of this expression agree Avith what I had computed sev- 

 eral years ago, by following Clauraut's method, published in the Memoires de 

 l\icademie des Sciences^ Sfc. FariSy 1748^ 



tThe same maybe observed of Newton's method given in the Princijna Lib, 

 1. Prop, xiv, cor. 2. Where the motion of the moon from Apogee to Perigee is 



put equal to 180° I J — , C being equal to I m' of the above notation. Sub- 



1 — 4c 



stituting this value, and developing the expression according to the powers of 

 m*, the motion of the apsides in one revolution from apogee to apogee would be 

 expressed by | m* . [1 + V m^J.SGO' of which the first term agrees with the 

 formula (r), and is therefore correct, but the other term is inaccurate. 



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