382 



Prof. H. Xagaoka on Magnetic 



For the first coil we have to put the value of \ in (9'), 

 and for the second coil k 2 in (9") for calculating K and E ; 

 the value of tj is positive for the first coil, but negative for 

 the second. This can be easily distinguished by adding the 

 suffix + or — to X and Y. Thus we may conveniently 

 write for these components 



X = X + + X_; Y=Y + + Y_. 



(12) 



We shall in the first place confine our attention to the 

 evaluation of the values of X and Y for the single coil. 

 Evidently /c, K, and K— E can be expressed by means 

 of ^-functions as given in every text-book on elliptic 

 functions, so that by expansion in terms of q we obtain 

 easily 



jpE-2(K-E) j = 19277^1(1 + 20^ + 225^+...); 



pE = 327^1(1 + 30 ? 2 + 423f/ 4 + 3986? 6 +...).^ 



These expressions are of great importance in calculating 

 the value of the magnetic force. Thus for a single coil 



X= 967r--^- ? l(l + 20^ + 2255 4 +...) 



vap 



Y = 16w-— L. #j^(l + 30o 2 + 423o 4 +3986 ? 6 +...) ! 



-6q(l + 20q 2 + 225q 4 +...) 1 : 

 and for a double coil referred to the middle point as origin, 



(13) 



96 



-=- q\ (1 1- 20? 2 ± + 225<£ + 



Y ± = 16tt -~-=q\ j^ (1 + 30^ + 423^ + 3986^+- .. 



-6? ± (l + 20? 2 ± + 225? 4 ± 4- ... 

 Putting #=f, y = rj in (8), we find for the case of 



y (IL; 



single coil 



k 2 = 



4af 



(a + £)» + 9*' 



