170 Proceedings of the Royal Society 
If we represent by AA', ^'a' by A' A", then 
also rja + 7) a' = ^AA" = A"A^ , 
namely at right angles to AA", and of equal length. To construct 
COS W 
A"Ai we describe the circle passing through the three points 
sin?f? ^ 
A, A', and A", and the point D where A"A^ meets the circumfer- 
ence will give 
nrf\ co^io -ttta 
A D = — A A, . 
sin V) 
This is because the angle ADA" is supplementary to the angle 
AA'A", which itself is supplementary to w; and as the triangle 
AA"D is rectangular in A" by construction, we have, for the length 
AAj = AA", 
A D = cot AA, = — (-na + r? a ) , 
^ sin w^' ' ’ 
