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



wards from A and inwards towards B. These two systems of 

 lines subdivide the whole surface of the conducting sheet into 

 quadrilaterals, such as P Q and Q R. The actual direction of 

 the flow at any point P, due to the combined action of the source 

 and sink, will evidently be intermediate between the directions 

 of the primary streams AP and PB through the same point; 

 similarly the direction of the resultant current at Q will be in- 

 termediate between the direction of the primary streams A Q 

 and Q B ; in other words, the line of resultant flow through P 

 will enter the quadrilateral P Q at P, and the line of resultant 

 flow through Q will pass at Q from the quadrilateral P Q into 

 the quadrilateral Q B. And it can be shown that the points P, 

 Q, and B are points on the same line of flow — that is, that a 

 continuous curve can be drawn through P, Q, and B such that 

 no electricity flows across it. Thus, let the points of intersection 

 of the primary flow-line AP with the primary flow-lines through 

 B which are nearest to P B on either side of it, be marked P ; 

 and P ; respectively ; and similarly let Q' and Q ; , R' and R y be 

 the points where A Q and A R intersect the next flow-lines on 

 either side of Q B and B B respectively. Then, since the total 

 flow of electricity from A between the lines A P and A Q is 

 everywhere the same and is equal to the flow towards B between 

 the lines P B and Q B, which is likewise everywhere the same, 

 the quantity of electricity flowing in a given time across any of 

 the lines P Q', P y Q, P P ; , Q' Q, which bound the quadrilateral 

 P Q, is equal. Consequently, considering either of the triangles 

 P Q' Q or P P y Q, the flow inwards across P Q' or P P y is equal 

 to the flow outwards across Q' Q or P y Q; and therefore there 

 cannot on the whole be any flow across a line drawn inside the 

 quadrilateral from P to Q. Evidently also, by drawing a suffi- 

 cient number of straight lines through A in directions inter- 

 mediate between A P and A Q, and an equal number of lines 

 through B in directions lying between P B and Q B, keeping 

 the angle between consecutive lines constant in both cases, we 

 can subdivide the line P Q into portions as short as we please, 

 and such that no electricity flows across any of them. Hence 

 P and Q are points on the same line of flow; and it follows, 

 similarly, that the point B is also situated on this line. The 

 same kind of reasoning also proves that a second line of flow 

 passes through the points P', Q', and B', and a third through 

 the points P y , Q y , and R r Moreover, since the strength of the 

 flow between the lines P' Q' B' and P Q B is measured by the 

 quantity of electricity which in unit of time crosses any of the 

 lines P'P, PQ', Q'Q, QR', or R' B, and the strength of the 

 flow between the lines P Q R and P y Q y R, is measured by the 

 quantity which similarly crosses any of the lines P P y , P y Q, Q Q y , 



