104 PROPERTIES OF CIRCULAR CURRENTS. 



To within factors, the terms of correction are the same /i', f" 

 as those of the component X at the point A given by equation (29). 

 The factors of /[' and /j' are moreover composed of y and S, as 

 these functions themselves are of a and x that is to say, as functions 

 in respect of a circular current of radius y, parallel to the former for 

 a given point, the abscissa of which would be 8. 



If a be the angle which the magnetic axis A 2 A : of the needle 

 makes with the plane of the current, which gives 



8 = / sin a, 



y = l cos a, 

 we get 



D = 47I/;' / cos a i +-~(i- 5 sin 3 a) +^ (1-14 sin 2 a + 21 sin 4 a) . 

 L /o /o J 



If we replace the functions /J,', /i', / 2 ' by their values (736), 

 we get finally, 



a 2 r i a 2 



(47) D = 27 r-2/co S 



(2.4) 



\/ 4 ~| 



sin 4 a 1 . 

 /w 4 J 



When x is made = 0, that is, when the middle of the needle is at 

 the centre of the frame, the formula reduces to 



(48) D = ^ 2 /cosa i+A(i_ 5 sin 2 a)^- 2 



3 2 5 / V 4 1 



+ . ' xo (1-14 sin 2 a + 2 1 sin 4 a ) . 

 ( 2 . 4 ) 2 \ /0 4 J 



The components f and rj of the force </> will be calculated 

 in the same way, but this force is very small, and has no interest 

 in experiments ; when a coil is made to act on a magnet suspended 

 by a wire, the only effect of the force <j> is to deflect the wire a little 

 from the vertical. 



747. When the coil has a frame with a rectangular section, the 



a 2 

 principal term 271- must be replaced by the value of G (12). As to 



the terms of correction within the bracket, it will be sufficient to take 

 for the quantities a and u, their values relative to the near circum- 

 ference of the coil, unless the magnet has an appreciable length as 

 compared with the diameter of the frame. 



