286 Prof. Norton on Molecular Physics, 



, mdz 1*, 



have —s- — 9 ; and for the action of nr, 



I 



mdz m 1 ,z ~ 

 ^-—^-tan-^-fC. 



y'+z* y y 



For the entire action of nb, we have the definite integral 

 -tan -1 -* In a similar manner we obtain for the opposing 



action of d, 



m u 

 ,tan -1 - 



y + d y + d 



The effective impulsive action of the portion nodb of the mag- 

 net will then be 



m , ,u m , . u 

 — tan" 1 ,tan -1 - 



y y y+d y+d 



The effective action of the other portion, nao c, of the magnet 



will be 



m .v m , , v 

 — tan -1 7 tan -1 • 



y y y+d y+d 



We therefore have for the entire action of the magnet 



mi, ,u v\ m ( u v \ 

 s= -(tan- 1 - -ftan- 1 -)— A tan -1 ; + tan -1 ;), 



y\ y y) y+d\ y+d y+dy 



w 

 or 



w= 7 ^^Ycafb)-~- d {aYCcfd). . . (a) 

 When y is large as compared with d, we have approximately 



772 



w= ~ (arc afc+ arc bfd) (b) 



To obtain the equation of the curve of equal impulsive force, 

 let mn=x, and rm=t. Then nr=z=x + t, and 



nb = u = x H- mb = x + a. 



Also na = v = a—x. Hence 



,/. ,x + a , ,«— x\ m ( x + a ,a—x\ 



(tan- 1 - Man- 1 ) -^( tan" 1 —, + tan" 1 ->) = C.(c 



\ y y J y + d\ y-td y—dj x 



m, 



y 



C here represents the constant intensity of the impulsive force 

 of the magnet for one curve. The value of C decreases as the 

 distance of the curve from m (fig. 9) increases. The equation (b) 

 shows that for the larger curves, except near the magnet, afc + bfd 

 must vary in nearly the same ratio with the ordinate y, from one 

 point to another of the curve. To the left of the line a k the 



