X sin/5+smce 

 hence 



268.] ATTRACTION. 163 



n^; (8) 



2 



' (9) 

 i.e. the attraction at P bisects the angle APB subtended at P by the rod. 



267. These results might have been derived from the problem of 

 Art. 265. For it is easy to see that the attraction exerted at P by the 

 straight rod AB is the same as that exerted by the circular arc ab (Fig, 

 79) described about P with radius/ and bounded by PA, PB. This will 

 follow if it can be shown that the attraction at P of the element QQ 1 ' 

 of the rod is equal to that of the element qq* of the arc contained 

 between the same radii vectores. Now the attraction of QQ' is 

 K P'QQ7 r ~> while that of W* is K P '$$'//' Projecting QQ' on the 

 circle of radius r, we have 



n' _p 



QQ'cosO~~r 



or since the triangle POQ gives cosO=fl/r, 



which proves the proposition. 



268. It has been shown in Art. 266 that the attraction at any point 

 P exerted by a straight rod AB bisects the angle APB ; it is therefore 

 tangent to the hyperbola passing through P and having A, B as its foci. 

 Hence if in any plane through AB the system of confocal hyperbolas 

 be constructed with A, B as foci, the direction of the attraction at any 

 point P in the plane is along the tangent to the hyperbola that passes 

 through P. These hyperbolas having everywhere the direction of the 

 resulting attractive force, are called the lines of force. 



An ellipse passing through P and having the same foci A, B would 

 have the bisector of the angle APB as its normal. The confocal 

 ellipses about A, B as foci form the so-called orthogonal system of the 

 lines of force. If such an ellipse be regarded as offering a normal 

 resistance, the point P would be kept in equilibrium under the action 

 of the attraction of the rod and the reaction of the curve. The con- 

 focal ellipses are therefore called equilibrium, or level, lines, or also 

 for a reason that will appear later equipotential lines. 



