444 Mr. S. Butter worth on the Coefficients of 



3. Equal Circles. 



Take the radius of either circle as the unit of: length and 

 let x be the distance of their planes. Then when x is large 

 and the circles are coaxial, the mutual inductance is * 



, T 2tt 2 /. 3 25 245 , \ n . 



the multiplying factor to obtain the nth term from the 

 preceding term being 



1 /2rc — A 2 n-1 



^ ,2 \ n / n + 1 



Hence, using spherical coordinates with the centre of the 

 fixed circle as origin and its axis as the axis of coordinates, 

 when the centre of the moving circle is at r, the mutual 

 inductance is 



M 



_2tt_ 2 / 3P4 25Pe 245 Pa, \ ( . 



- r z \ r * 4 r 2 + 24 r 4 ~ 128 r 6 + " " 7' w 



in which P ?l = P TC (cos#) is the zonal harmonic of order n. 



When the circles are coplanar, 0=-^-, and (2) then 

 becomes 



■M ^Vl ■ 9 125 8575 L \ IVL\ 



the multiplying factor to obtain the nth term being 



1 / 2n-l \ 3 ?i-l 

 2r 2 \ n ) w + 1" 



Formulae (1), (2), and (3) converge if r>2, but the 

 convergence is rather slow if r is less than 3. 



When x is small and the circles are coaxial the mutual 

 induction is t 



0= =47r{x -2 + f^ 2 (V I) - S( X °-§) 



-(^-S)----}' (4) 



M 



L " lb V " 



35 



in which X = log< 



£ 



To obtain the mutual induction for non-coaxial circles 



* Havelock, Phil. Mag-, xv. p. 332 (1908). 



t Coffin, Bull. Bureau of Standards, ii. p. 113 (1906) ; Havelock, 

 loc. cit. 



