260 Proceedings of the Eoyal Iriah Academy. 



And proceeding in like manner with this result, we shall obtain 



tan"^ «i + tan~^ a^ f tan"^ a^ + tan"^ «4 = tan~^ , 



I - ^2 + ri 



where «i, a.,, a^, a^ are roots of 



x^ - ')\ x^ + r^x'^ - r-i X \ rij^ = o. 

 ISTow assume 



tan"^ «i + tan^^ a^ + tan~^ ffg + . . . + tan"^ a„ = tan~' ~ ^ °- — ^-^, 



I - Co ^ e^^- . . . 



where «i, «2, «3, • • • «„ are roots of 



x^ - Ci x"''^ + Co x'"-' - C3 x"-^ +...+(- I )" <?„ = 0. 



Here it will be necessary to distinguish the cases when n is even 

 and when n is odd. 



( I ) If ^^ be even, the last term in the numerator of the right-hand 



member of the equation is (- i ) 2 . c„_i, and the last term in the de- 



nominator is (- 1)2 . c„. Hence, adding on tan~' «„+i to both sides of 

 the equation, we shall get 



tan"^^! + tan"^ a^ + tavr^ a-^ + . . . + tan"^ «„ + tan""" «„+i = tan~^ — , 



jD 



where 



A- = {Ci + ff„+i) - (^3 + C^ «„+i) + {C^ + Ci ff„+i) - ... 4- (- I 2 '((.„_! + C„_3 . «„+i) 



n 



and ^ = I - {co + ci ff,,.+i) + {c^ + c^ a„+-^) --... + (-1)2 (c„ + c,,_i . «„+i)- 

 But «i, «2, «3, • • . fi!,., «„+i are roots of 



(2.-" - Ci x"~^ + ^o x^'^ - ... - c„_i . *• + c„) (it; - fi«„+i) = 0, 

 that is, of 



.^j^+i _ (ci + «„^i) a;" + (^2 + (?! (j„+i) d;«-i - (^3 ^ Co a„+i) . a;"-- + . . . 



Hence, since n is even, 



tan"' ^1 + tan"' «2 + tan"' «3 + . . . + tan"' «„ + tan"' «,,+! 



= tan-' '^1 ~ ^3 + '^5 - • • • + (- 0^ ~ ' • C i + (- 0^ • 4+1 



I -to^ti-...\{- \Y .i,. 



