of Electricity in a uniform plane conducting Surface,. 459 



obtained in § 11 in the following form, 



ZAPB=rca, 



where n is a parameter whose value changes by unity on passing 

 from one line to the next, and a is the angle between adjacent 

 rays of the equiangular pencil of flow-lines produced by either 

 pole separately. If the straight line AB be taken as the axis of 

 w, the above equation may be written 



ZAPB=ZPBX-ZPAX=rca. 



Moreover it was shown, in § 10, that the system of flow-lines 

 resulting from the composition of any two similar systems is 

 obtained by drawing lines through the alternate angles of the 

 quadrilaterals produced by the mutual intersection of the lines 

 of the component systems, in directions concurrent with both the 

 flow-lines which intersect each other at each angle. Hence, if the 

 poles A and B (PI. IX. fig. 1) were of the same sign (both sources 

 or both sinks), the lines of flow of the resultant system, instead 

 of passing through the points P, Q, B and P„ Q 2 , Rj, &c, as 

 there represented, would pass through the supplementary angles 

 of the quadrilaterals, namely from Q' to Pj, from R/ to Q p &c. 

 It is evident that, as we pass from point to point of a line so 

 drawn, the angles which the radii vectores from the poles make 

 with the axis of x change equally, but in opposite directions - 9 

 hence this equation may be written 



ZPBX + ZPAX=rca. 



28. If, instead of the line through the poles, any other straight 

 line in the plane of flow be taken as the direction of reference 

 for measuring the angles made by the radii vectores, the form in 

 which the equations to the lines of flow have just been given will 

 still be applicable. Thus, if the line X (PL X. fig. 1), making any 

 angle $ with A B, be taken as the direction of reference, and if a 

 and b be the points where it is met by P A and P B respectively, it 

 is clear that the angles P a X and P b X are each of them less than 

 the previous angles PAX and P B X by the angle S, but that 

 for each flow-line they give a constant sum or difference accord- 

 ing as the poles are of the same or of opposite kinds. Hence, 

 denoting the angles which the radii vectores make with any fixed 

 line by 6 l and 6 2 respectively, and agreeing to distinguish sources 

 and sinks by a difference of intrinsic sign of the angles corre- 

 sponding to them, we may write the equation to the flow-lines 

 due to two equal poles, whether similar or opposite, in the 

 general form 



Oi + e^nu. 



29. In like manner, the lines of flow due to any number of 



