394 Messrs. G. C. Foster and O. J. Lodge on the Flow of 



lines close to each pole are the same in all respects as they would 

 be if the other pole were absent. 



13. The circular form of the flow-lines for the case we are 

 considering can be demonstrated in various other ways. We will 

 give here one additional proof, on account of its great simplicity. 



The flow through any point P due to a source at A and a sink 

 at B, being the resultant of the currents through the same point 

 due to A and B taken separately, will be represented in strength 

 and direction by the third side of a triangle whose other two 

 sides represent the currents from A and towards B, respectively, 

 in strength and direction. But the strengths of the two com- 

 ponent currents are inversely as the distances A P and B P re- 

 spectively (§ 4) : hence the following construction (fig. 2). 

 From B draw B T parallel to A P, and make its length a third 

 proportional to A P and P B ; then P T gives the direction of 

 the flow at P, and its length is proportional to the strength of 

 the current at P. The similarity of the triangles A P B and 

 P B T gives the angle B P T equal to the angle P A B ; and con- 

 sequently (converse of Euclid, III. 32) the locus of P is a circle 

 through A and B. 



14. It was shown in § 10 how a system of lines of flow can 

 be traced out by successive points. To be able to draw them 

 continuously with compasses we only require to know the posi- 

 tion of the centres ; and these are easily found from the following 

 considerations. Since the circles of which the flow-lines are arcs 

 pass through the poles A and B, their centres lie in the straight 

 line at right angles to A B, through O its middle point. If C 

 (fig. 3) be the centre of the circle which gives the flow-line 

 through any point P, the angle at C is equal to the angle at P 

 — the angle characteristic of the given flow-line ; and therefore 

 the angle A C is the complement of the angle at P. Putting 

 AB=2#, we have 



OC = «.tanOAC = 0.tan(!--APBV 



Let the number of lines to be drawn be m, then the constant 

 difference between the angles contained under consecutive lines 



will be 



2tt 



— =u > 

 m 



and the several lines will be given by making the angle at the 

 circumference successively equal to 



0, a, 2a..., ir—u, 7r, 7r + «,... 27r— 2a, 27T — a, 2ir, 



or, what is the same thing, to 



0, a, 2a, ...7T— a, — 7T, — {it — a), , . . — 2a, —a, 0, 



