261 Froceeding^ of the Royal Irish Acade})iy. ■ 



Cor. — Putting r/i = tanai, r/2 = tana2, . . . r/„ = tan a,^, it f ollo-vvs that 



111 - 7?3+ 7?5- ... 00 



tan (ai + Oo + as + . . . + a„) = 



I -h^h,~ 



"where h,. denotes the sum of the homogeneous products of tan ai, tan a.,, 

 tan as, . . . tan a,„ and their powers of r dimensions — a result of a like 

 type to the trigonometrical one already noticed, and is thus established 

 ■without the use of De Moivres theorem. 



4. Putting «! = % = ^3 = . . . = (In = X = tan 0, in the preceding result, 

 ■we may deduce 



n tan ^x = tan" 



^hiX - h^x^ + h^x 



I -hoX- + hiX^ - . . . oc ' 

 and 



hi tan e - h; tan^^ + h, tan^O - . . . oo 



tan nO 



I - h.2 tan-^ + 7?4 tan*^ - . . . oo ' 



•where h^ now denotes the numher of homogeneous products of n things 

 of r dimensions, i.e. \n + r- i -^ \n - i \r. 



And further, if a; = i , we have 



hi - 7^3 + 7^3 - ... cc ^ nir 



—— = tan — , 



I - /^a -f /<4 - . . . cc 4 



)2— I 



which = (- I ) 2 if « be odd, and = o or oo according as n is of form 4;^ 

 or 4«j + 2. 



Thus hi - 7«3 4- 7?5 - . . . 00 = o, if n be of form 4/^, 

 and I - ho -\- hi - . . . x = o, \i n be of form 4«t + 2. 



SUPPLEMENT. 



I . The theorem 



^ hi - A3 + 7(5 - ... 00 



tan ( ai + a., + a. + . . . + a„ i = =■ 



I - ho + hi- ... 00 



may also be obtained in the following way : — We have 

 (cos ai + \/- I sin ai)(cos a^ + V - i sin a.,) . . . (cos a„ + v - i ) sin a,;) 

 = cos (tti + a-j + tts + . . . + a„) + V - I sin (aj -r ao + a;. + . . . + a„). 



