of Edinburgh, Session 1869-70. 95 
= PD 2 is also a line of flow. In this case both circles are neces- 
sarily real. 
It is clearly impossible that the same system should have two 
pairs of circular stream lines of either of the classes we have 
analysed. Nor can two complete pairs of different classes occur, 
since otherwise two stream lines would intersect. But three real 
and an imaginary circle are possible, if ABCD lie on a circle, 
and at the same time obey the condition for a pair of circles of 
the second class, that is, if AB produced pass through the pole of 
CD with respect to the circle ABCD. The three circles are mani- 
festly orthogonal, and their radical centre is centre of the fourth 
(imaginary) circle. 
If the circle through ABCD is 
S = x 2 + y 2 - a 2 = 0 , 
the lines AB, CD respectively 
u — hx + ky - a 2 = 0 
v = h'x + h'y — a 2 = 0 , 
we have 
hli -j- hk' — a? — 0 , 
and the second and third circles become 
S - 2u = 0 
S - Zv = 0 . 
The fourth or imaginary circle is 
S - 2w= 0, 
where 
w __ a*(V-k)x + af(h-h r )y _ ^ 
hk‘ — kh’ hk' — kh' 
w = 0 representing the polar of the intersection of AB, CD. 
Thus the equation to the stream lines may be written 
(S - 2t*)(S - 2v) + AS(S - 2w) = 0 , 
or, 
(1 + A)S 2 - 2(m + v + \w ) S + 4 uv = 0 , 
which degenerates into a cubic when A. = - 1. 
The equations may, in general, be simplified by a proper choice 
of co-ordinates. 
