Mr. R. Moon on the Equation of Laplace's Coefficients. 439 

 whence, proceeding as before, we find 



2n-lb n _ l 



n • n+ 1 — n — 1 n— 2 



2n — 36 n _3— w — 2 ?i — 3c„ 



n .n+l— n — Sn — 4 



_ 2tj — 56„- 5 — 7& — 4?z — 5g M _ 4 



(5) 



/I . 7Z+1— ft — 5 72 — 6 



&c. &c. 



Hence, putting T for log e ( tan 2^ and ^M for ff^l dun • %M * 

 we have the following expression for o>, viz. 



a) = (^^sT| n + ^_ 2 cos^; w - 2 +&c.) ./ ^i(T+* ^-1) 1 



L+^ 2 (T~^V-I)J 



+ (^_ 1 c^ ra - 1 + ^_ 3 c^s^| w - 3 4-&c.).f ^(T + <£^=Tn 



+(c n _ 2 ^s~9r 2 +^ 4 c^- 4 + &c.) . I *a t +* ^5P \ 



L+f 2 "(T-^^/-l)J 

 &c. &c., 



where, if we choose to put a n = 1, as we may do, we shall have 

 a n =1, 



n . 7i— 1 



#tt-2 — 



72. 71+1—71 — 1 71 — 2 



ft . Ti—1 ft— 2ft— 3 



(rc.ft+1-ft— lft— 2)(ft.ft + l— ft— 3ti— 4)' 



a n -e= — 



n— In — 2 Ti — 371 — 471 — 5 



(ft.ft + 1 — 7i — lft— 2)(ft. 7i+ 1 — n — 3ft — 4)(ft.ft + l — 7i — 5?i— 6 

 &c. &c, 



and where the constants 6„_i, b n - 3} &c, c n _ 2 > e»-4> &c., taken in 

 order, are determined by equations (4) and (5), &c. in terms of 

 known quantities. 



