ACTION OF A COIL ON A MAGNETIC NEEDLE. 103 



of the needle, y and x + 8 the co-ordinates of the pole A v y and 

 x - 8 those of the pole A 2 ; we shall assume that these poles are 

 equal to unity. 



Let X, Tj, and X 2 , Y 2 denote the components of the field of the 

 frame in absolute values at the points A l and A 2 . As the poles A z 

 and A 2 are of opposite signs, the values of the components f and f\ 

 of the force, and the moment D of the couple are 



(YL-Y,,) 3. 



On the other hand, if X and Y are the components of the field at 

 the point A, the co-ordinates of which are x and y, we have 



1.2 a* 8 1.2.3 ^ i. 2. 3. 4 <h; 4 



t)jc 1.2 (to 2 1.2.3 ^r 1-2.3.4 for 4 



The values of X 2 and Y 2 will be deduced from the preceding by 

 changing the sign 8 ; it follows that 



T 8 2 ^) 2 X 8 yx "I 

 = 2 X + -^ + - + ... 

 2 ^ 2 24 c)* 4 J 



If as above we put (736) 



; + 2 / 2 / +...), 



limiting the expansion to terms of the fourth order, we shall have 



+/;y + (/; - sy:')- + (/; - 8/7) - 



The properties (31) of the functions f^fi ---- give directly 



- 1 2/6 2 + 85 2 )]. 



