54 LECTURES AND ESSAYS [1869- 



straight lines, forming equally inclined rays through that 

 point. 



The condition for a rectilineal branch is in general 

 that the sources must be either on the line or be two by 

 two, each other s images on the line. For if not, remove 

 all the sources on the line and all pairs of sources which 

 are each other s images in the line. Next, remove all 

 sources on one side of the line by placing equal sources of 

 opposite sign at the place of their images. The straight 

 line is still a stream line, and on one side of it there are no 

 sources, and therefore constant potential, which is absurd. 

 Similarly it can be shown that a circle is a possible stream 

 line only when the sources are on the curve or image each 

 other. From this it follows that no finite number of 

 sources can give parallel rectilineal streams or non- 

 intersecting circular streams. 



A similar investigation applies to equipotential lines. 

 The image of a point in a rectilineal equipotential line is 

 the same in position as the image in a stream line, but of 

 opposite sign. No source can lie on an equipotential line. 

 Hence, to show that for right equipotential line the points 

 must image two by two, we have only to remove all 

 sources on one side of the line, placing equal sources of the 

 same sign at their images. The line is still equipotential, 

 therefore we may suppose it charged to constant potential, 

 and all sources removed. Hence all stream lines become 

 rectilineal, which is absurd. Similarly if a circle is equi 

 potential, the sources must balance about it two by two, 

 i.e. must be in a straight line with its centre, at distances 

 to which the radius is mean proportional otherwise we 

 can find a system reducible to a single point at the centre 

 of the circle, and in which all stream lines are rectilineal. 

 Hence, no incomplete system can have a rectilineal or 

 circular potential line. 



Points of Inflexion occur at all points on the locus 



d 2 u d 2 u dy d-u ~dy* 



T-O + 2 T- j- -/ + j-o :r- = . 17 . 



dx~ dxdy dx dy 2 dx 



