190 Mr J. Brill, On Conjugate Functions [Feb. 27, 
This gives us w=cosh™(z/a), i.e. 2=acosh w; and, if we turn the 
plane of w through a right angle by writing tw for w, we arrive at 
the well-known form 
2 = acos WwW, 
Let C be the centre of a circle, P a point on the circle, and 
S and §’ two inverse points with respect to the circle. Then, 
taking the above figure, we have the angular relation 
PCA = PSC + CPS= PSC + PS'C. 
Further, if we draw two circles passing through P and having 
their centres at S and 8S’, the tangents at P to the three circles 
in the figure will be respectively perpendicular to SP, S'P and CP. 
Thus the relation between the angles which these three tangents 
make with the line S’SA is exactly similar to that given above. 
Thus we see that a system of coaxal circles, having S and S’ for 
limiting points, will form an equipotential system, being derivable 
with the aid of our theorem from two systems of concentric circles 
having S and S’ for centres. To obtain the function of a complex 
variable from which this system is derivable, we have to solve the 
equation 
eee tito (2— a) — log (2 + a) + log k, 
” dz 
where a and & are real quantities. This gives us 
w= ue log = 
2a Peta’ 
We will now make & =a, so that this becomes 
2w = log eae 
2+a 
which may be written in the equivalent form z=—acoth w. 
And, if we turn the plane of w and the plane of z each through a 
right angle by writing 7w for w and 2z for z, we have the form 
z2=acotw, 
