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



16. Equipotential Lines. — The flow-lines for the case of one 

 source and one equal sink having been determined, the form of 

 the equipotential lines is at once given by the consideration that 

 the two sets of lines cut each other orthogonally (§ 2) ; and it is 

 a well-known geometrical result that the system of lines ortho- 

 gonal to the system of circles which, as has been seen, represents 

 the flow-lines for this case, is another system of circles having 

 their centres on the line through the poles, each circle cutting 

 this line once internally and once externally in points situated 

 harmonically with respect to its extremities. The simplest 

 general expression for such a system of circles is the equation 



r 



?=* 



where c is a quantity which remains constant for each circle but 

 varies from each circle to the next, and r and r 1 are the distances 

 from any point of the curve to the source and sink respectively. 

 These general properties of the equipotential lines are easily 

 established by referring to the construction employed in § 13. 

 We there saw that P T (fig. 2) is a tangent to the line of flow 

 through P ; and consequently it is normal to the equipotential 

 line through the same point. If we produce P T to cut A B 

 produced in C, we have the triangles BCP and P C A similar, 

 and 



CP 2 =CA.CB; 



whence it appears that if tangents to the lines of flow be drawn 

 from any point C in A B produced, the distance from C to the 

 points where these tangents touch the lines of flow is constant 

 and depends only on the distances of the point C from A and B 

 respectively. Therefore, if a circle be drawn with centre C and 

 radius C P, it cuts all the lines of flow at right angles, and is 

 consequently an equipotential line. If it is only required to find 

 the centre of the equipotential circle through any point P, the 

 simplest method is to make an angle B P C equal to the angle 

 BAP; then the point where P C and A B intersect is the centre 

 to be found. 



The similarity of the triangles BCP and P C A also gives 



AP_r_AC 

 PB~V~CP ; 



or the ratio of the radii vectores from the two poles is constant 

 for a given circle, as already stated. 



17. The above method suffices for drawing the equipotential 

 lines through any number of given points, but not for placing 

 them systematically (or so that the difference of potential between 

 consecutive lines may be constant). For this purpose we may 



