i87s] STREAM LINES 61 



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 A B C D 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*+y*-a* = 0, 

 the lines AB, CD respectively 



u = hx + ky - a 2 = 



v = h x + k y - a 2 = 0, 

 we have 



hh + kk -a*=Q, 



and the second and third circles become 



S - 2u = 

 S - 2v - 0. 



The fourth or imaginary circle is 



S - 2W = 0, 



where 



* g (fr-*)*a 8 (*-* / )y - 

 hk -kh &quot; hk -kti 



w = representing the polar of the intersection of AB, CD. 

 Thus the equation to the stream lines may be written 



(S - 2U) (S - 2V) + AS(S - 2W) = 0, 



or, 



(i + A)S 2 - 2(u + v + \w)S + 4uv = 0, 



which degenerates into a cubic when A = - i. 



The equations may, in general, be simplified by a 

 proper choice of co-ordinates. 



Take, for example, the case when S - 2u, S - zv are 

 equal circles. 



Then A 2 + k 2 = W + k&quot; 2 , 



