AxGLiN — Trigonometrical Notes. 259 



XVII. — Trigoxometeical Notes. Ey A, H. Aj^glin, M. A., 

 P.R.S.E., &c. 



[Read, May 26, 1884.] 

 To prove that 



tan"^ «! + tan"^ % + tan"^ a^-]- . . . + tan~' a„ = tair ' — ^ s • • • 



I - <>2 + ^4 - . . . 



where «i, a^, a^, . . . a,, are roots of the equation 



«" - Cix"^'^ + c-iX""'^ - c^x"^^ + . . . + (- I )"'<;,j = ; 

 and hence to establish the well-knoAvn trioonometjical formula 



tan (aj + aj + tta + . . . + a„) = 



I - Co + t^i - . . 



"where c,. denotes the sum. of the products of tan aj, tan a^, tan a^, . . , 

 tan a„, taken r at a time, wUhoid the use of the symhol v - i . 



A + B 



I . Since tan~^^ + tan~^i5 = tan"^ — ^, 



1 — A.1) 



we have tan"^rt!i + tan~'«o = tan^^ —^ , 



where a^, a^ are roots of the equation x^ - f^x ^- fn = ^ . 

 Adding on tan"V<3 to both sides of this result, we get 



tan"^«i + x-xnr^(h + tan"'«3 = tan"^ r. 



But (?i, Un, f/3 are roots of 



{x- - p^x + p2){oc - ttz) = o, 



that is, of 01? - {pi -\- a^x^ + {ih + piaz)x - p%az = 0. 

 Hence tan"^ «, + tan"^ ^3 + tan"^ a-^ = tan"^ -^ , 



1-^2 



where <?i, ^2, a^ are roots of 



x^ - qiX- + qoX ~ q^ - o. 



K. I, A. PROC, SER. 11., VOL. IV. SCIENCE. 



2F 



