228 



du being the change of position produced by a single element 

 of the railing ; so that we have 



Xdu = ^-=^3 {sin a + 3 sin (a - 2^) dx} . 



X = a tan 0, dx = ^^ ^ , D 



2£)3 

 But 



ad(l> T\ _ ^ 

 cos^^ cos^ 



and substituting, and integrating, 



Xu = 9-1 ( sin a + 3 sin (a - 2^) j cos (j)dijt 



= — {4cos (a-0) + COS (a -3^) -cos(a + ^)}, 

 and between the limits ^ = and ^ = j3, 



^^^ " S { ^ ^^" (° " 2 ) ^^° 2 ^ ^^" (a - /3) sin 2 /3 j . 



The value of mn in this expression was obtained, as before 

 stated, by observing the effect of a single piece of railing in 

 a known position. For convenience of calculation, this piece 

 was placed upon the perpendicular let fall from the centre of 

 the magnet upon the line of the rails, the bars being parallel 

 to their original position. In this position 0=0; and there- 

 fore the moment of the force exerted by a single bar, at the 



distance D, is 



2mm' . 

 -n^ sm a ; 



so that, if £ denote the disturbance produced by a portion of 

 the railing, whose length is unity, in this position, 



^^ 2mn . 

 Xe = -TTT- sma. 



Finally, dividing the former result by this, mn and X are 

 eliminated, and we have 



Dh 



4a^sina 



I 4 sin fa - y) sin^ + sin (a - |3) sin 2/3 | 



