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



equal poles are represented by the equation 2,0= net, the angles 

 which the radii vectores from the several poles make with any 

 fixed line being respectively denoted by 6 lf 6 2J . . . • In order 

 to prove this, it is sufficient to show that, if the expression holds 

 good for any given number of poles, it will still be true for one 

 pole more ; for since the formula has been proved for two poles, 

 this would enable us to advance, by successive additions of one 

 pole at a time, to any number. 



Let, then, the curves P"Q'R, FQI^, andPQ,R 2 (Pl.X.fig.2) 

 represent parts of three successive lines of flow due to any number 

 k of equal poles, and let them be characterized respectively 

 by the following values of 2,6, namely (n + l)a, not, and (n— l)a. 

 If there is another pole of the same strength at L, which we will 

 suppose in the first instance to be a source, the pencil of flow- 

 lines diverging from it with the common difference of angle a 

 will intersect the flow-lines from the k sources, producing a 

 number of quadrilaterals through whose opposite angles the 

 lines of flow will pass which result from the action of the pole 

 at L combined with that of all the rest (§ 10). If the direction 

 of the flow along the lines of the component systems be as re- 

 presented by the arrow-heads* in the figure, one line of the 

 resultant system will pass through the points P' and Q', another 

 through P, Q, and R, and a third through Q x and R r Let the 

 lines joining the source L with the points P, Q, and R cut the 

 fixed line X in p, q, and r respectively ; then, for the flow-line 

 through Q due to the k poles, we have by hypothesis 



and for the line through the same point due to the source L 

 we have 



/.Q,qX = ma, 



and therefore, for both together, 



20-fQgfX=(w-r-m)a; 

 Por the point P, we have similarly 



26 + VpX. = {n-l)a + {m + l)a; 



* For the purpose of the argument in the text, we may assume arbi- 

 trarily either the direction ofjlow along the assumed system of lines of 

 flow, or the direction of increase in the value of n (in no), but not both ; 

 for a definite assumption regarding one of these conditions determines the 

 other also. If the system of flow-lines for any given set of poles is built 

 up by successive repetitions for one pole at a time of the construction given 

 in § 10, it will be found that the conventional distinction between positive 

 and negative angles leads to the following rule as to the connexion between 

 the two above-named conditions : — Going along a given flow-line with the 

 flow, any line on the left hand corresponds to a higher value of n than the 

 given line, while a line on the right hand corresponds to a lower value of n. 



