438 Mr. R. Moon on the Equation of Laplace's Coefficients. 



from which the law of formation of the terms in the series for 

 A m or A n is obvious. These being known, A n _! can be deter- 

 mined by means of the equation 



A^-ifsintf^p +n. Z+Tf sin BAn^d&y 



or 



«A n _ 



= 2A„+ sin 0-^p- +n .n+lfsin0 A n _ x dd ; . . (3) 



for, from what has preceded, it is evident that we may assume 



A n _ 1 ==^_ 1 co75] w - 1 + ^_ 3 c^sl| n - 3 + ^_ 5 -c^s?| n - 5 +&c. 



(where b n - Xi b n ^ 3 , &c. are constants) ; and substituting this value 

 in (3), we shall get 



0=2a n cos0\ n + 2a n - 2 cos~0| n - 2 + 2a„_ 4 cos0|"~ 4 + &c. 



\n— 4 



+ ?»-l^_ 1 cos6'P + n-3^_ 3 cos6'r- 2 + n-5^ 5 cos6>| n - 4 + &c. 

 -n^lbn^ cosTr- 2 -^3^_ 3 co75| ra " 4 -&c. 



— n>n+ l^-i 



cos 0\* 



—n.n + lb n - 3 



cos 



e\ 



»-2 



n.n+lb n . 



cos 01 



n n-2 " """"^-4 



whence, equating to zero the coefficients of the different powers of 

 cos 0, we derive the following : 



2na n 



■&c. 



#«_! = 



n . ?i + l— n , n — 1 



a __ 2n — 2a n _ 2 —n— 1 n— 2 b n _ x 



O n -S ., ., = y 



b n -b — 



2n— 4<a n _ 4 — n— 3 n— 4 # n _ 



rc. w+1 — » — 4w- -5 



(4) 



&c. 



In like manner, if we assume 



in- 2 



A n _ 2 =c w _ 2 cos 0\ n - 2 + c n -tCos 0f +*«-« cos 0|*- 8 + &c, 



may determine the coefficients c n -. 2 > c n ~ 4t &c. by means 

 ; equation 



A n ., = -i^siu^-^ 2 - +w. ^+T J s i n A n _ 2 d6 \ 



or 



0=2A n - x + sm6--^ + n .n + l §sin0\ n _. 2 d0 ; 



