58 LECTURES AND ESSAYS [1869- 



If the vertical angle is less than -, the rectilineal branch 



is the transverse axis ; if greater than -, it is the conjugate. 



If the points are all in a line, the vertices of the hyperbola 

 lie on that line, and are the points of zero flow, which are 

 easily found. If one point is half way between the other 

 two, we have two rectilineal branches and two hyperbolas, 

 the conjugate axis of the one being equal to the transverse 

 axis of the other. The hyperbolas are, therefore, confocal. 



Four Points. Complete System. 



Singular Points. If A and B are sources, C and D 

 sinks, there is a singular point at P, if the circles APC, 

 BPD, and also APD, BPC touch at P. Hence, there are 

 no real singular points if the sides of the quadrilateral 

 ACBD intersect, unless all the points be on a circle, 

 which in this case contains all the singular points. 



Straight Lines. The one stream line which has as 

 asymptote is of the third degree. If a straight line is one 

 factor, the other factor is a conic, which is always a circle. 

 For if A, C are the images of B, D respectively in the 

 straight line, a circle can be drawn through them, which 

 is obviously the branch sought. But if A, B lie without 

 the line, and C, D on it, a circle through A, B having its 

 centre O in CD produced, so that OA is a mean propor 

 tional between OC and OD is the circle required. If 

 ABCD are all on a straight line, the other branch is 

 manifestly a circle with centre on the line. 



Conies. The parabola is an impossible conic for any 

 finite number of points. For the parabola has two 

 asymptotes meeting at infinity. Hence the centre of 

 gravity of an incomplete system, or of the sinks and 

 sources separately in a complete system, must be at an 

 infinite distance, which is absurd. The conies are there 

 fore central. 



The hyperbola, which has two asymptotes, is only 



