266 Proceedings of the Royal Irish Academy. 



Again, 

 cos nB + v/- I sin nO = (cos 6 - v^- i sin ^)"" = sec" ^ ( i - v - i tan 6)'". 

 Proceeding as before, we find 

 cos nO Gos'^6 = I -/?2 tan^^ + h^ ioxi^O - . . . oc = A, suppose, 

 sin 7i6 cos"' 6 = hi tan $ - h^ tan^^ + h^ tan^^ - . . . oo = C, suppose, 

 where h^ = \n + r - i i- \n - i jr. 



Thus we deduce the following relations : — 

 AB = cos-fiO, 



^ = (cos^r, 



CD = sin- nO, 



- = (cos^)-; 



and hence 



4 C 

 ^ = ~, or AD = BC, 



AB + CB = I, 



CD = ABiim'ne, 

 AfABCD = sin" inO, 

 AC=BBeos"'e. 



