388 Magnetic Field of Circular Currents. 



The deviations of the second order vanish for the double 

 coils here considered, so that we have only to discuss those 

 of the fourth order upward. The beauty of the arrangement 

 lies chiefly in the elimination of this most important deviation, 

 and we at once notice how terms of higher order are insigni- 

 ficant and make the use of the arrangement for obtaining 

 uniform field very important in some practical applications. 



In the following, deviations of fourth order for X and Y 

 components are given in tables. 



Deviation of fourth order for X-component : — 



l^(Vf-3^ 3 ) = c, . - . (24) (Fig. 7.) 



Asvmptotes : +40° 53' 48". 



Nonnals at the vertices : ± 20° 58' 32"; + 66° 7' 18". 



Deviation of fourth order for Y-component : — 



-T2%(3r-24£V + 8^) = c. . (25) (Fig. 8.) 



Asymptotes : ±19° 52' 30" ; ±59° 26' 40". 

 Normals at the vertices: 0°; 180°; ±90°; 



±40° 53' 48". 



The comparison of the figures 7 and 8 with the corre- 

 sponding deviations of fourth order for a single coil in 

 figures 3 and 4 shows how the deviations in double coils 

 are small, and especially the gradients. The use of double 

 coils in exact measurements of magnetic field, as for 

 example of the terrestrial magnetic force, is strongly to be 

 recommended. 



The discussion here given refers to single layer coils. By 

 winding the coils in different layers, w r e have to choose 

 a proper section of winding layers. For a rectangular 

 section, Maxwell shows that when the ratio of the depth 

 to the breadth is as 6 to 5*57, an important correction due to 

 the finite section of the winding is made to disappear. 



As before discussed, the deviations of magnetic force 

 due to terms of different orders have an analogy to the 

 aberrations of lenses and mirrors, and will be ol great 

 advantage in designing galvanometers and other electric 

 instruments, in which high order of accuracy and the 

 uniformity of the field are a principal aim. 



My thanks are especially^ due to Mr. S. Sakurai, assistant 

 in the Institute for Physical and Chemical Research, for 

 verifying the formulae and for constructing the diagrams 

 of deviations of different orders. 



Physical Institute, 



Imperial Universitv, Tokyo, 

 July 1920. 



