262 Proceedings of the Royal Irish Academy. 



2. The following deductioiis from the foregoing general theorem 

 may be noticed : — 



( I ) If ^1 = ^2 = ^3 = ...=^,^ = X, 



CiX -Cc^a? + c^x^ - . . . 

 then n tan"' x = tan~^ — 



1 - CzOp' ^- CiX^ - . . . 



•where c^ now denotes the number of combinations of n things taken r 

 at a time, /. e. I « ^ \ r \n- r ; and if x = tan 0, 



^ c, tan e-c^ Uw^e + ^5 tan^^ - . . . 

 tan nB = 



1 - Co tan-t^ + Ci tan^^ - 

 (2) Further, if .<- = 1, we get 



mr Ci - ^3 + fg 

 tan — = 



4 1 -Co-\- Ci - . 

 If n be odd, we haA'e 



71—1 



tan — = (- I ) 2 ; 



« — I 



4 



1 - c, + fi - . . . + (- I ) 2 . C„_l 



Mnd if n be even, 



n 



^ = or oc , 



n 



I - Co + fi - . . . + (- I ) 2 . C„ 



according as n is of form 4/^ or A^m 4 2. Thus 



n 



Cj _ C3 4 C5 - . . . + (- I ) ^ ' • ^"-1 = °' ^* " ^*^ °^ ^°^'™ 4"' ' 



w 



and I - Co 4 C4 - . . . + (- I )2 • <?n = o, if ?? be of form \m + 2. 



3. An interesting theorem of a like nature to the preceding one 

 may appropriately be noticed here. 

 To show that 



/?i - 7?3 + 7^5 - . . . 00 

 tan-' ffi + tan-' r/., + tan'' c/3 + . . . 4 tan"' a„ = tan — y j , 



1 — /?o + /'i — • • • °C 



where h,. is the sum of the homogeneous products of </i, a-z, (ii> ■ ■ • c»y 

 aud their powers all of r dimensions. 



