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



is true for the lines of flow produced by any number of equal 

 poles, whether of the same sign or not. 



Further, if we have poles of unequal strengths supplying (or 

 removing) quantities of electricity in the unit of time denoted, 

 say, by Q r , Q 2 , . . ., these may be regarded as each of them pro- 

 duced by the coalescence of a corresponding number of equal 

 poles supplying in unit of time a quantity of electricity q which 

 is taken small enough to be a common measure of all the quan- 

 tities Q„ Q 2 , .... In applying the general formula to such a 

 case, the value of 6 corresponding to each pole would have to be 

 taken a number of times equal to the number of constituent 

 poles of strength q required to make up the actual pole. 

 Hence 



2(Q<9)=Q 1 1 + QA+---=™< . . / (15) 

 is a perfectly general expression for the lines of flow produced 

 by any number of poles of any strength. 



31. Equipotential Lines.-— It was shown in § 17 that the sys- 

 tem of equipotential lines resulting from the composition of any 

 two systems, for which the constant difference of potential on 

 passing from one line to another is the same, is given by draw- 

 ing lines through the alternate angles of the quadrilaterals 

 formed by the mutual intersection of the lines of the component 

 systems, if the angles are taken in such order that on going 

 from any one to the next we pass, in one of the component 

 systems, to a line of higher, and in the other to one of lower 

 potential. We know also (see § 5) that the equipotential lines 

 due to a single pole are concentric circles whose radii vary ac- 

 cording to the terms of a geometrical progression, the potential 

 increasing in the case of a source as the radii decrease, and, in 

 the case of a sink, as the radii increase. Hence it follows 

 very simply, by the application of reasoning exactly analogous, 

 to that employed in § 29, that if P be a point on an equi- 

 potential line due to any number of equal sources at the points 

 A, B, C, . . . ., and sinks at the points A', B', . . . ., 

 AP . BP . CP . . . . „ 

 AT . B'P . . . . "^ constant ' 

 where fi is the constant ratio of the radii of consecutive equi- 

 potential circles due to a single pole, while n is a number cha- 

 racteristic of the particular line on which the point P is situ- 

 ated and increasing by unity if this point passes from any given 

 line to the line of next lower potential. For shortness we may 

 put r x for AP, r 2 for BP, . . . ., and write the above equation 

 thus, 



J — /* > 



r 



