1873] STREAM LINES 59 



possible when the cubic reduces to a conic. This demands 

 that the centre of gravity of sinks and sources shall 

 coincide, i.e. that AB, CD are diagonals of a parallelo 

 gram. The asymptotes must meet at right angles, and the 

 hyperbola is equilateral. It is obvious, indeed, that in 

 this case the sources and sinks give separately sets of 

 concentric rectangular hyperbolas, of which the one 

 passing through the four points belongs to both sets, and 

 is the only asymptotic curve of the complete system. 



In this case the equipotential lines are lemniscates. 

 Let the origin be the centre of the system, 20, and 2b the 

 diagonals of the parallelogram, a and /3 their angles with 

 the initial line. At any point P 



AP 2 . BP 2 + ACP 2 . DP 2 = 0. 

 That is, 



r* + a* - 2aW cos 2 - a + A(r 4 + fr - 2&V 2 cos 2 d - /3) = 



(i + A) (r 4 + 4 ) - 2f 2 cos 20 (a 2 cos 2a + Afr 2 cos 2/3) 



+ 2r 2 sin 20 (a 2 sin 2a + Afr 2 sin 2/3) = 0. 



W i a 2 sin2a 



Wn en A = - -- -, the curve becomes 



(b 2 sin 2/3 - a 2 sin 2a) (r 4 + a 4 ) - 2a?b 2 r 2 sin 2(y8 - a) cos 20 = 0, 



a lemniscate, with foci on the initial line, and centre at the 

 origin. 



If the parallelogram is a rectangle a = b, and the curve is 



H - 20V 2 COSj L_! cos 20 + a 4 - 0. 



COS /3 + a 



It is easily shown that the stream lines orthogonal to 

 these are lemniscates with the same centre, passing 

 through the four points, one of which becomes a circle 

 when the parallelogram is rectangular. 



The ellipse appears to be an impossible conic for four 

 points, for conies occur in pairs orthogonal to each other. 

 The orthogonal of the ellipse must be a confocal hyperbola, 

 which is impossible, the only hyperbola being that dis- 



