352 PLUCKER ON THE THEORY OF DIAMAGNETISM. 



this way we find, when we neglect the factor — ^~- and assume 

 c as the unit of length, 



— 5{cos^.r^+(cos^ <jb— 3)r^ + cos</) .r}(r— cose/)) 

 -f {3 cos </> . 7-2 + 2(cos2 <^— 3)r + cos </)} (r^— 2r cos <^ + 1) = 0, 



or when we develope 



2 cos<^ ./•'* + (4 cos^<^— 9)r^— cos^(cos^</)— 4)r^ 



— (5cos20-6)r~cos</)=O (12) 



This is the polar equation of the curve sought. To express 

 the curve by means of rectangular co-ordinates, we will multiply 

 by Vy r cos <^ by x, and for r^ substitute {x^-'ry'^). We then find 

 (2a7-9)(^^+^2)2 + (4^H4^ + 6)(2/2 + ^2) 



—a?(.2?2 + 5.^4-1) =0 (13) 



The figure shows the course of the curve, which in every 

 position of the lever gives the point, whose moment of rotation 

 is a maximum. This, less strongly pronounced, curve is of the 

 fifth degree, and consists of two separate portions. One of these 

 forms an oval which passes through the point P and cuts the 

 axial line a second time in the point S, which is half the distance 

 of the pole P from the centre O. The point of this oval corre- 

 sponds to an attraction. The other portion of the curve, like the 

 geometric locus of the indifferent points, has the point P as 

 double point, and an asymptote which is at right angles to the 

 axial line, but which stands further by one-half from the centre 

 of rotation, and an oval which lies within the oval of the geo- 

 metric locus just mentioned. The points of this second portion 

 of the curve correspond to a repulsion on the part of the 

 pole P. 



To obtain the total action of both poles on a lever of given 

 magnitude, which consists of an infinite number of infinitely 

 small iron bars, all perpendicular to the direction of the length, 

 the expressions (6) and (7) must be multiplied by dr and then 

 integrated. Let the length of the lever be 2R. For the sake 

 of brevity, let 



_ /*'' 1:^ .|r2(cos2 <^-3) + (rM ^)cos <^"l=^, 



^0 (r2--2rcos</)-f l)n J 



r 1^ ^/r2(cos2<i-3)-(/^ + r)cos<f)|=^. 



^ (rH2rcos</)-M)n ^ ' ^ j tj 



