120 PARTICULAR CASES OF EQUILIBRIUM. 



on each unit of length of the cylinder S, and a layer - m on each unit 

 of length of the cylinder S', will replace the action of the two 

 unlimited lines A and A' (61) for all points between the two sur- 

 faces ; the figure will correspond, in this case, to the problem of a 

 condenser formed of two unlimited excentric cylinders. 



137. Let us suppose that the distance za approximates to zero, 

 but that the density A varies so that the product 20 A remains con- 

 stant. The potential at the distance r, in a direction which makes 

 the angle o> with the straight line, will be 



\ 



- 



\ r ] 



This equation represents circumferences whose radii vary as 

 the reciprocals of the series of even numbers. 



We have in like manner for the lines of force, 



2a sn to 



<D to = - 



Thus the radii of the circumferences which represent the lines of 

 force vary also as the reciprocals of the series of even numbers. 



138. SYSTEMS OF REVOLUTION. To determine the sections of 

 elementary tubes of force on an equipotential surface, we shall take 

 on the one hand equidistant meridian planes, and on the other, 

 points placed on the meridian section so that in the revolution 

 about an axis, they divide the surfaces into successive zones, corres- 

 ponding to the same flow. The surface will thus be divided into 

 curvilinear rectangles corresponding to the same flow, which will 

 be taken equal to unity. 



139. A uniform field may always be considered as one of 

 revolution about any line parallel to the direction of the force ; we 

 may therefore apply to it this mode of representation. An equipo- 

 tential surface, which is a plane perpendicular to an axis, will be 

 intersected by a series of circumferences comprising between them 

 zones of constant surface. The radii, increasing according to the 

 same law as Newton's rings, will be proportional to the square roots 

 of consecutive numbers. The lines of force will then be represented 

 in the meridian plane by right lines, parallel to the axis, and whose 

 distances from the axis are as the square roots of consecutive whole 

 numbers. 



If F is the strength of the field, and w the angle of the two 



