314 Mr J. Brill, On the Geometrical Interpretation of the [Nov. 26, 
AL, AF BE BSF 
are respectively 
e+iy=a—d+i(b+e), 
e—ty=atd+zi(b—-c), 
xe+iy=at+d—2(b-0o), 
x—ty=a—d—i(b+ ce). 
The coordinates of the point of intersection of AZ and BJ are 
z=a-—d, y=b+e, 
and those of the point of intersection of AJ and BI are 
we=atd, y=—(b-c). 
The remaining points of intersection of the lines AJ, AJ, BI, BJ 
are in general imaginary, as also are the intersections of these 
lines with those obtained by joining a second similar pair with 
the circular points at infinity. 
Call the two real points of intersection P and Q; and denote 
the coordinates of the imaginary points A and B by the symbols 
(a,, 8,) and (a@,, 8,). Let z, and z, be the complex quantities 
that correspond to the pomts P and @ according to Gauss’s 
method of representing complex quantities by pomts. Then 
we have 
2,=a—ad+i(b+e)=a+b+it(e+id)=2,+78,, 
and similarly 
z,=a+d—2(b-—c)=a—w+ i (c—rd) =a, + 28,. 
5. Suppose that we have an equipotential system of curves 
given by the equation -=/f(w) =f (E+%). Also let 
F(E+%m) = $(E, 0) + OP (E, 9); 
x= (&7) and y=¥(é, 7). 
To find the points in which the curve 7=@ 1s cut by the con- 
secutive curve 7 = a+ da, we have the equation 
f (E+ ts) =f {E+ t (a +da)} 
=f (E+ ta) +idaf’ (E+ 2a). 
so that 
Thus the requisite values of € are given by the equation 
f (E+%a)=0. 
Now let X+%u be a root of the equation 7’ (w)=0. Then we 
have 
E+ia=A+H+imM=A47 (w— a) +a, 
