466 Messrs, G. C. Foster and 0. J. Lodge on the Flow 



is > - — one branch encloses each pole : and when s is < - — , 

 2ira r 2ira 



one branch surrounds both poles and the other surrounds neither, 

 being within the central loop of the self-cutting curve. Each 

 of the loci cuts any straight line through the origin harmonically 

 with respect to its intersection with the circle r = « and to the 

 origin. 



Three equal poles.- — A special case coming under this head, 

 which admits of some important applications, is that in which 

 there are three equal poles in a straight line, one of the outside 

 ones being of opposite sign to the other two — for instance, two 

 sources A and A', and one sink C. In this case the flow-line 

 given by 6 1 + 6 2 — 3 = 7r consists of two branches, one of which 

 is the flow-line passing through the poles ; while the other is a 

 circle whose centre is the sink C (fig. 5), and whose radius C P is a 

 mean proportional between. the distances from the sink to the two 

 sources respectively (or CP= '/CA . CA'). If the two sources 

 coincide, the systems of lines of flow and equipotential lines 

 become what are shown in fig. 6, and the circular branch of the 

 flow-line net =ir passes through the pole of double strength. In 

 accordance with what was said above (§ 34), every flow-line of 

 the system passes twice through the double source and once 

 through the sink. 



. Four equal poles.— -Four equal poles of the same sign situated 

 at the corners of a square give the system of flow-lines repre- 

 sented in fig. 7. The two diagonals of the square and the twa 

 straight lines through the middle point parallel to the sides of 

 the square are given equally by the values not — it and ft<x=0. 

 For net = Jtt we get a curve of four branches, one of which is 

 situated symmetrically in each quadrant. 



A special case of four equal poles, which is important in con- 

 sequence of its being susceptible of experimental verification, is 

 presented by a combination of two sources and two sinks arranged 

 in a manner that may be regarded as a duplication, with inver- 

 sion of signs, of the system of three poles shown in fig. 5. Let 

 the pole at C be a sink, those at A and A' being sources. Let 

 two additional sinks be placed at B and B' (fig. 8), points on 

 another straight line through C, and let a source equal to them 

 be put at C ; then the three poles of this new system would give 

 by themselves, as one of their flow-lines, the straight line C B 

 and the circle with centre C and radius = \/CB . CB'. If, how- 

 ever, the three new poles be made equal to. the three poles of 

 the first set, the sink originally at C and the equal source now 

 put there will exactly compensate each other, and the combined 

 system will be reduced to two sources at A and A' and two sinks 



